The two division properties of exponents: the quotient of powers rule, which subtracts exponents, and the power of a quotient rule, which distributes the exponent over numerator and denominator. Includes negative results rewritten with positive exponents, checking by cancelling factors, and comparing quantities by ratio.
Subject: Algebra 1 · 65 slides · symbolic lesson
Open the interactive version of this deck
Title
Algebra 1 · Chapter 8 — Exponents and Exponential Functions
Division Properties of Exponents
Objectives
Five outcomes, each one you can test yourself on with a pencil and no answer key.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.4 Division Properties of Exponents §8.4, pp. 462-468 — the lesson these objectives are drawn from
Warm-up
Lesson 8.1 found that multiplying powers adds their exponents. Division undoes multiplication, so this lesson asks what it does to exponents.
Discussion prompt
Write four to the fifth over four cubed as individual factors and cancel. What single power is left?
Hint: Three factors on the bottom cancel three on the top.
Answer:
\[ \dfrac{4^5}{4^3} = \dfrac{4 \cdot 4 \cdot 4 \cdot 4 \cdot 4}{4 \cdot 4 \cdot 4} = 4^2 \]
Three factors cancel and two are left, so the exponents subtracted. Adding for multiplication and subtracting for division is the same relationship those two operations always have.
Concept
To divide powers that have the same base, subtract the exponents. The rule holds because the factors of the denominator cancel against factors of the numerator, leaving the difference.
quotient of powers property — To divide powers with the same base, subtract the exponents: a to the m over a to the n is a to the m minus n, for a nonzero.
The base must be nonzero, since dividing by zero is undefined.
Figure (svg): The quotient of powers property stated with an example
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.4 Division Properties of Exponents §8.4, pp. 462-462
Section
Section 1
Concept
Dividing powers with the same base subtracts the exponents, because the denominator's factors cancel against the numerator's and the difference is what remains.
\[ \dfrac{a^m}{a^n} = a^{m - n}, \quad a \neq 0 \]
The bases must match, exactly as they did for the product rule.
Figure (svg): A quotient of powers written out and cancelled factor by factor
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.4 Division Properties of Exponents §8.4, pp. 462-462 — the quotient of powers property and Example 1
Picture it
Five on top, three below.
Figure (svg): A quotient of powers written out and cancelled factor by factor
Nothing here needed a rule: the cancelling is ordinary fraction arithmetic. The rule is a shortcut for a computation you could always have done.
Worked example
This is Example 1 from the textbook.
\[ \text{Simplify } \; \dfrac{6^5}{6^4} \; \text{ and } \; \dfrac{y^3}{y^5}. \]
Take the first
Why: Subtract four from five.
\[ 6 ^{1} \]
Evaluate
Why: Six to the first is six.
\[ 6 \]
Take the second
Why: Subtract five from three.
\[ y ^{-2} \]
Rewrite with a positive exponent
Why: The reciprocal of y squared.
\[ 1 / y ^{2} \]
Figure (svg): A quotient whose exponents subtract to a negative number
\[ 6 \qquad \dfrac{1}{y^2} \]
Verify: check the second by cancelling
Why: Three y's on top and five below leaves two y's in the denominator, which is one over y squared. The Study Tip in the textbook points out exactly this: cancelling common factors gives the same answer without using the rule at all.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.4 Division Properties of Exponents §8.4, pp. 462-462
Sorting
The rule needs matching bases.
Sort into buckets
Sort each quotient by whether the quotient of powers rule applies.
The matching-base requirement is the same one the product rule had, and for the same reason: cancelling needs identical factors.
Worked example
Guided Practice 1 to 4. One gives a zero exponent.
\[ \text{Simplify } \; \dfrac{8^8}{8^6}, \quad \dfrac{(-3)^3}{(-3)^2}, \quad \dfrac{x^4}{x^4}, \quad \dfrac{a^5}{a^9}. \]
Take the first
Why: Eight minus six.
\[ 8 ^{2} = 64 \]
Take the second
Why: Three minus two, keeping the negative base.
\[ (-3) ^{1} = -3 \]
Take the third
Why: Four minus four is zero.
\[ x ^{0} = 1 \]
Take the fourth
Why: Five minus nine is negative four.
\[ 1 / a ^{4} \]
Figure (svg): The solution to Worked example four from guided practice shown as a ladder of expressions, one row per algebraic move
\[ 64, \quad -3, \quad 1, \quad \dfrac{1}{a^4} \]
Verify: check the third without the rule
Why: Anything divided by itself is one, provided it is not zero — and x to the fourth over x to the fourth is exactly that. The rule gives x to the zero, which Lesson 8.2 defined as one, so the two routes agree as they must.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.4 Division Properties of Exponents §8.4, pp. 462-462
Trap
\[ \dfrac{6^6}{6^3} \]
Divide the exponents to get 6^2
Why: The expression is a division, so dividing the exponents feels like the matching operation.
Writing it out gives six factors over three, which cancels to three factors — so the answer is six cubed. The two happen to differ here, and on other numbers they would differ wildly.
\[ \dfrac{6^6}{6^3} = 6^{6-3} = 6^3 \]
Subtract the exponents, because the factors cancel
Why: The division is between the powers, and cancelling removes a run of factors.
Testing on two to the fourth over two squared settles it: sixteen over four is four, which is two squared rather than two.
Faded example
Numerator's exponent minus the denominator's.
Fill in the blanks
\dfrac41 = 6^___}} = 6^___} = 6
Why: Four factors cancel and one remains, so the answer is six to the first, which is six. Subtracting in the other order would give six to the negative one, or a sixth, which is the reciprocal of the truth.
Elimination
Simplify a to the seventh over a cubed.
Eliminate the wrong options
Which is correct?
Survives elimination: A
Why: Seven factors on top and three below leaves four on top, so the numerator's exponent comes first in the subtraction. Testing with a equal to two settles it: one hundred and twenty-eight over eight is sixteen, which is two to the fourth.
Socratic
Multiplying added.
Discussion prompt
Explain why dividing powers with the same base subtracts their exponents, using what happens to the factors. Then say why that is the operation you would expect, given what multiplication did.
Hint: Ask what cancelling removes.
Answer:
Each factor in the denominator cancels one in the numerator, so a run of n factors is removed from a run of m. What is left is m minus n factors, which is what the exponent on the answer records.
Division undoes multiplication, so whatever multiplication does to exponents, division should undo. Multiplying added, so dividing subtracts — and that relationship means the two rules are really one fact seen from two sides, which is worth noticing rather than memorising them apart.
Section
Section 2
Concept
The subtraction may give zero or a negative number. A zero exponent gives one and a negative exponent gives a reciprocal, so the definitions from Lesson 8.2 complete the simplification.
A simplified answer is conventionally written with positive exponents.
Figure (svg): A quotient whose exponents subtract to a negative number
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.4 Division Properties of Exponents §8.4, pp. 462-463 — Example 1(b) and the Study Tip on using only positive exponents
Picture it
Subtract, then reciprocate.
Figure (svg): A quotient whose exponents subtract to a negative number
Both forms are correct answers. The reciprocal form is preferred because a simplified expression is expected to show only positive exponents.
Worked example
This is Example 1(b) from the textbook.
\[ \text{Simplify } \; \dfrac{y^3}{y^5} \; \text{ with a positive exponent.} \]
Apply the rule
Why: Three minus five.
\[ y ^{-2} \]
Recognise the negative exponent
Why: It means a reciprocal, from Lesson 8.2.
Rewrite it
Why: One over y squared.
\[ 1 / y ^{2} \]
Note the convention
Why: A simplified answer shows only positive exponents.
Figure (svg): The same quotient simplified by the rule and by cancelling
\[ \dfrac{1}{y^2} \]
Verify: confirm by cancelling
Why: Three y's on top and five below leaves two below, giving one over y squared. Cancelling uses no rule at all, so it is a genuinely independent check on the subtraction.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.4 Division Properties of Exponents §8.4, pp. 462-462
Sorting
Look at the sign of the difference.
Sort into buckets
Sort each quotient by the form of its simplified answer.
The three cases correspond exactly to whether the top exponent is larger, equal or smaller. Comparing them before subtracting predicts the shape of the answer.
Worked example
The two runs of factors cancel completely.
\[ \text{Simplify } \; \dfrac{x^4}{x^4} \; \text{ two ways.} \]
Apply the rule
Why: Four minus four is zero.
\[ x ^{0} \]
Use the definition
Why: A nonzero base to the zero power is one.
\[ 1 \]
Check without the rule
Why: Anything nonzero divided by itself is one.
\[ 1 \]
Note the condition
Why: x must not be zero, or the quotient is undefined.
\[ x\text{ not } 0 \]
Figure (svg): The solution to Worked example a difference of zero shown as a ladder of expressions, one row per algebraic move
\[ \dfrac{x^4}{x^4} = x^0 = 1 \]
Verify: say why the two routes had to agree
Why: The rule was defined in Lesson 8.2 precisely so that results like this would come out right. That a quotient of equal quantities is one was known long before the definition, and the definition was chosen to match it.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.4 Division Properties of Exponents §8.4, pp. 462-462
Error analysis
The student simplified two quotients of powers.
Annotate
On: \( \begin{aligned} \frac{y^3}{y^5} &= y^2 \\ \frac{x^4}{x^4} &= 0 \end{aligned} \)
Cancelling the factors is available whenever the rule is in doubt, and it uses nothing but ordinary fraction arithmetic.
Faded example
A negative exponent reciprocates.
Fill in the blanks
\dfrac-44 = a^___ = a^___} = \dfrac______}}}
Why: The difference of negative four becomes a fourth power in the denominator. Both forms are correct and the second is the simplified one, since a simplified expression shows only positive exponents.
Elimination
The exponents are equal.
Eliminate the wrong options
Which is the value?
Survives elimination: A
Why: Everything cancels, so the value is one — and the restriction on x is genuine, since the original expression has a denominator. Option D is worth noticing: the answer is one wherever the expression is defined, and it is not defined at zero.
Socratic
The negative form is correct too.
Discussion prompt
Say why a simplified answer is conventionally written without negative exponents, and whether anything mathematical is at stake. Then say when you might deliberately leave a negative exponent in place.
Hint: Ask what a convention is for.
Answer:
Nothing mathematical is at stake — y to the negative two and one over y squared are the same number for every allowed y. The convention exists so that two people simplifying the same expression arrive at the same written form, which makes answers comparable and marking possible.
You would leave negative exponents in place when they make a pattern visible, as in scientific notation where ten to the negative four is far clearer than one over ten thousand, or when a further calculation is about to use the exponent rules again. Lesson 8.5 is built entirely on that convenience.
Section
Section 3
Concept
To find a power of a quotient, raise the numerator and the denominator to that power and then divide. The exponent distributes because multiplying fractions multiplies the parts separately.
\[ \left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}, \quad b \neq 0 \]
This is the power of a product rule applied to a fraction.
Figure (svg): A quotient raised to a power, expanded and regrouped
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.4 Division Properties of Exponents §8.4, pp. 463-463 — the power of a quotient property and Example 2
Picture it
Four copies, then regroup.
Figure (svg): A quotient raised to a power, expanded and regrouped
The numerators gather and the denominators gather, exactly as the factors did in Lesson 8.1's power of a product rule. A quotient is a product with a reciprocal in it.
Worked example
This is Example 2 from the textbook.
\[ \text{Simplify } \; \left(\tfrac{3}{2}\right)^2, \quad \left(\tfrac{y}{3}\right)^3, \quad \left(\tfrac{7}{4}\right)^{-3}. \]
Take the first
Why: Three squared over two squared.
\[ \frac{9}{4} \]
Take the second
Why: y cubed over three cubed.
\[ y ^{3} / 27 \]
Take the third
Why: Seven to the negative three over four to the negative three.
Simplify the third
Why: The reciprocals swap top and bottom.
\[ \frac{64}{343} \]
Figure (svg): A quotient raised to a power, expanded and regrouped
\[ \tfrac{9}{4}, \quad \dfrac{y^3}{27}, \quad \tfrac{64}{343} \]
Verify: check the third a different way
Why: A negative exponent on a fraction flips it, so seven quarters to the negative three is four sevenths cubed, which is sixty-four over three hundred and forty-three. The two routes agree, and the flipping shortcut is worth knowing.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.4 Division Properties of Exponents §8.4, pp. 463-463
Faded example
Numerator and denominator.
Fill in the blanks
\left(\dfrac416\right)^4 = \dfrac______}}} = \dfrac______}
Why: The denominator receives the exponent along with the numerator, giving sixteen rather than two. Leaving the denominator unraised is the standard error with this rule and changes the answer by a factor of eight.
Worked example
Guided Practice 5 to 8. One has a variable denominator.
\[ \text{Simplify } \; \left(\tfrac{5}{4}\right)^3, \quad \left(\tfrac{x}{2}\right)^4, \quad \left(\tfrac{5}{3}\right)^{-2}, \quad \left(\tfrac{1}{x}\right)^5. \]
Take the first
Why: Five cubed over four cubed.
\[ \frac{125}{64} \]
Take the second
Why: x to the fourth over sixteen.
\[ x ^{4} / 16 \]
Take the third
Why: The negative exponent flips the fraction.
\[ \frac{9}{25} \]
Take the fourth
Why: One to the fifth over x to the fifth.
\[ 1 / x ^{5} \]
Figure (svg): The solution to Worked example four from guided practice shown as a ladder of expressions, one row per algebraic move
\[ \tfrac{125}{64}, \quad \dfrac{x^4}{16}, \quad \tfrac{9}{25}, \quad \dfrac{1}{x^5} \]
Verify: check the denominator of the second
Why: Two to the fourth is sixteen, not eight. Both parts of the fraction receive the exponent, and forgetting to raise the denominator's number is the standard slip with this rule.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.4 Division Properties of Exponents §8.4, pp. 463-463
Trap
\[ \left(\dfrac{x}{2}\right)^4 \]
Write x to the fourth over two
Why: The exponent looks like it belongs to the letter it is nearest.
The brackets enclose the whole fraction, so the denominator is raised as well. Two to the fourth is sixteen, so the answer is x to the fourth over sixteen — eight times smaller than the version written.
\[ \left(\dfrac{x}{2}\right)^4 = \dfrac{x^4}{2^4} = \dfrac{x^4}{16} \]
Raise both the numerator and the denominator
Why: The rule distributes the exponent over the whole fraction.
Substituting x equal to two settles it: one squared to the fourth is one, and the wrong version gives eight.
Prediction
The whole fraction is raised to a negative power.
Predict first
What is (5/3)^(-2)?
Correct: 9/25, since the fraction flips and is then squared.
\[ \left(\tfrac{5}{3}\right)^{-2} = \left(\tfrac{3}{5}\right)^{2} = \tfrac{9}{25} \]
Why: The negative exponent reciprocates the whole fraction, giving three fifths, and squaring that gives nine over twenty-five. The negative sign never reaches the value's sign, as Lesson 8.2 established — and flipping the fraction is the quickest way to handle a negative exponent on a quotient.
Elimination
Simplify the quantity 2a over b, cubed.
Eliminate the wrong options
Which is correct?
Survives elimination: A
Why: Every factor of the numerator and the denominator receives the exponent, so two cubed is eight. This combines the power of a quotient rule with the power of a product rule, which is common and needs both applied.
Socratic
Lesson 8.1 had a rule for products.
Discussion prompt
Explain why the power of a quotient rule follows from the power of a product rule. Then say which lesson's definition makes the connection precise.
Hint: A quotient is a product with a reciprocal.
Answer:
Dividing by b is multiplying by b to the negative one, so a over b is the product of a and b to the negative one. Raising that product to a power distributes by Lesson 8.1's rule, giving a to the n times b to the negative n — which is a to the n over b to the n.
Lesson 8.2's definition of a negative exponent is what makes the rewriting legitimate, and Lesson 8.1's power of a power rule turns b to the negative one, raised to n, into b to the negative n. So this lesson's second rule is a consequence of three earlier ones rather than a new fact.
Section
Section 4
Concept
The chapter's rules now cover both operations. Multiplying powers adds exponents and dividing subtracts them; raising a power multiplies them, and raising a product or a quotient distributes.
Every rule concerns products and quotients; none applies to sums.
Figure (svg): The four exponent rules met so far
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.4 Division Properties of Exponents §8.4, pp. 463-463 — the Division Properties of Exponents summary alongside Lesson 8.1's
Picture it
Add, subtract, multiply, distribute.
Figure (svg): The four exponent rules met so far
The first two are a pair, since multiplying and dividing are inverse operations. Reading them together is easier than learning them apart.
Worked example
Naming each rule as it is used keeps the sequence straight.
\[ \text{Simplify } \; \dfrac{(2x^3)^2}{4x^5}. \]
Expand the numerator
Why: The power of a product rule squares both factors.
\[ 4 x ^{6} \]
Apply the power of a power rule
Why: Three times two is six.
\[ 4 x ^{6} \]
Divide the coefficients
Why: Four over four is one.
\[ 1 \]
Apply the quotient rule
Why: Six minus five is one.
Figure (svg): The solution to Worked example an expression needing three rules shown as a ladder of expressions, one row per algebraic move
\[ \dfrac{(2x^3)^2}{4x^5} = \dfrac{4x^6}{4x^5} = x \]
Verify: substitute a number
Why: At x equal to two the original is the quantity sixteen squared over one hundred and twenty-eight, which is two hundred and fifty-six over one hundred and twenty-eight, or two. The answer x gives two as well, confirming the whole chain.
Sorting
Read the structure.
Sort into buckets
Sort each expression by the rule that simplifies it.
The last two share a bucket because a quotient is a product in disguise, which is why one rule covers both.
Worked example
Reading the structure decides the action.
\[ \text{Which rule simplifies } \; a^3 a^4, \quad \dfrac{a^7}{a^2}, \quad (a^3)^4, \quad \left(\tfrac{a}{3}\right)^2? \]
Take the first
Why: Two powers multiplied.
Take the second
Why: Two powers divided.
Take the third
Why: A power raised again.
Take the fourth
Why: A quotient raised to a power.
Figure (svg): The four exponent rules met so far
\[ a^7, \quad a^5, \quad a^{12}, \quad \dfrac{a^2}{9} \]
Verify: notice how similar the four look
Why: All four involve the same letter and small exponents, and all four give different answers. Reading the structure rather than the numbers is what tells them apart, which is why naming the rule first is worth the second it takes.
Trap
\[ \dfrac{x^5 + x^3}{x^3} \]
Subtract the exponents to get x^2 + x^0, then simplify
Why: The expression looks like a quotient of powers, so the quotient rule seems to apply.
The numerator is a sum rather than a single power, so the rule does not apply directly. Factoring or splitting the fraction first is needed, and the answer here is x squared plus one.
\[ \dfrac{x^5 + x^3}{x^3} = \dfrac{x^5}{x^3} + \dfrac{x^3}{x^3} = x^2 + 1 \]
Split the fraction into separate quotients first
Why: Each term is divided by the denominator on its own.
Every rule in this chapter concerns products and quotients. A sum has to be broken into pieces before any of them can be used.
Faded example
Expand, then divide.
Fill in the blanks
\dfrac41 = \dfrac___x^6}___ = x^___}
Why: The numerator expands to four x to the sixth by two rules, and then the coefficients divide to one and the exponents subtract to one. Doing the numerator completely before dividing keeps the steps separate.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Multiplying powers | Dividing powers | |
|---|---|---|
| The operation on exponents | add | subtract |
| What happens to the factors | the runs are laid end to end | one run cancels the other |
| The answer compared with the first power | larger | smaller, and possibly a fraction |
The two rules are inverses of each other in every row, which is what you would expect from two inverse operations.
Socratic
Every rule here concerns products and quotients.
Discussion prompt
Explain why the exponent rules say nothing about a sum of powers. Then say what to do with a fraction whose numerator is a sum.
Hint: Ask what the rules are counting.
Answer:
All the rules come from counting factors, and a sum is not made of factors. Adding two powers produces a quantity that is not itself a power of anything, so there is no exponent to record and no rule to state.
A fraction with a sum on top can be split: each term is divided by the denominator separately, and then each of those quotients is a genuine quotient of powers. Factoring the numerator is the other route, and Chapter 10 develops it — but splitting is available immediately and needs nothing new.
Section
Section 5
Concept
One way to compare two numerical values is to look at their ratio. When both are written as powers, the quotient rules do the comparison in one step.
A ratio answers how many times, while a difference answers how much more.
Figure (svg): Two quantities compared by dividing rather than by subtracting
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.4 Division Properties of Exponents §8.4, pp. 462-468 — the chapter opener on baseball salaries and Exercise 59
Picture it
Divide rather than subtract.
Figure (svg): Two quantities compared by dividing rather than by subtracting
A ratio is scale-free, so it can compare quantities of very different sizes. That is why it is the natural comparison when powers are involved.
Worked example
This is the situation in Exercise 59 of the textbook.
\[ \text{A salary is about } 6 \times 10^5 \text{ in one year and } 4 \times 10^5 \text{ five years earlier. Compare them.} \]
Write the ratio
Why: The later over the earlier.
\[ \frac{6 x 10 ^{5}}{4 x 10 ^{5}} \]
Divide the coefficients
Why: Six over four is one and a half.
\[ 1.5 \]
Subtract the exponents
Why: Five minus five is zero.
\[ 10 ^{0} = 1 \]
State the comparison
Why: The later salary is one and a half times the earlier.
\[ 1.5 \times \]
Figure (svg): Two quantities compared by dividing rather than by subtracting
\[ \dfrac{6 \times 10^5}{4 \times 10^5} = 1.5 \]
Verify: compare with the difference instead
Why: Subtracting gives two hundred thousand dollars more, which is a different kind of answer. The ratio says fifty per cent more and the difference says two hundred thousand more — both true, and useful for different questions.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.4 Division Properties of Exponents §8.4, pp. 468-468
Faded example
Divide coefficients, subtract exponents.
Fill in the blanks
\dfrac43 = ___ \times 10^___ = 4 \times 10^___}
Why: The coefficients divide and the exponents subtract, giving four thousand. Doing the two parts separately is what makes comparisons of very large numbers manageable.
Worked example
The exponents do the heavy lifting.
\[ \text{Compare } \; 8 \times 10^7 \; \text{ with } \; 2 \times 10^4. \]
Divide the coefficients
Why: Eight over two is four.
\[ 4 \]
Subtract the exponents
Why: Seven minus four is three.
\[ 10 ^{3} \]
Combine
Why: Four times a thousand.
\[ 4000 \]
State the comparison
Why: The first is four thousand times the second.
\[ 4000 \times \]
Figure (svg): The solution to Worked example a ratio across different powers shown as a ladder of expressions, one row per algebraic move
\[ \dfrac{8 \times 10^7}{2 \times 10^4} = 4 \times 10^3 \]
Verify: check by writing both out
Why: Eighty million divided by twenty thousand is four thousand. Doing it with the full numbers agrees and takes far longer, which is exactly the saving the rules provide when the powers are large.
Trap
How many times larger is the later salary?
Subtract: 600,000 minus 400,000 is 200,000
Why: Subtracting is the familiar comparison, and it produces a definite number.
Two hundred thousand answers how much more, not how many times. The question asked for a ratio, which is one and a half.
\[ \dfrac{6 \times 10^5}{4 \times 10^5} = 1.5 \]
Read whether the question asks how much more or how many times
Why: The first is a difference and the second a ratio.
Both comparisons are legitimate and they answer different questions, so the wording decides which to compute.
Elimination
One quantity is nine million and another is three hundred.
Eliminate the wrong options
How many times larger is the first?
Survives elimination: A
Why: Nine over three is three, and ten to the sixth over ten squared is ten to the fourth, so the ratio is three times ten thousand. Both parts of each quantity contribute, and using only one of them is the standard slip.
Hypothesis
Predict before you decide.
Predict first
Which comparison is more useful for two salaries five years apart?
Correct: The ratio, because it says how much the salary grew proportionally.
This is why growth is usually reported as a percentage, which is a ratio in disguise.
Why: A ratio compares like with like regardless of scale, so a fifty per cent rise means the same whether the starting salary was small or large. The difference is a real and useful figure too, but it cannot be compared across different starting points — two hundred thousand is a huge rise on a small salary and a modest one on a large. The fourth option is simply false: differences are easy to compute at any size.
Socratic
The rules turn a division into a subtraction.
Discussion prompt
Explain why writing quantities as powers makes comparing them by ratio simple. Then say what that suggests about the notation Lesson 8.5 is about to introduce.
Hint: Ask what operation the exponents undergo.
Answer:
Dividing two powers subtracts their exponents, so a division of enormous numbers becomes a small subtraction. Eighty million over twenty thousand is a laborious long division and seven minus four is immediate, so the notation converts hard arithmetic into easy arithmetic.
That is exactly what scientific notation is for. Writing every number as a coefficient times a power of ten means multiplying and dividing them reduces to multiplying small coefficients and adding or subtracting exponents, which is the subject of Lesson 8.5 and the reason the notation exists at all.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Property | Looks like | Do this to the exponents |
|---|---|---|
| Product of powers | a^m x a^n | add them |
| Quotient of powers | a^m / a^n | subtract them |
| Power of a power | (a^m)^n | multiply them |
| Power of a quotient | (a/b)^n | give the exponent to top and bottom |
The first two rows are inverses of each other, as are multiplication and division themselves. That pairing halves what has to be remembered.
Pattern
Whether the expression divides, raises or both, the same five moves cover it.
Step five's cancelling check uses no rule at all, which makes it genuinely independent of whatever the rules produced.
Check
Numerator's exponent minus the denominator's.
Check your understanding
Simplify x^9 over x^4.
Answer: A
Why: Four factors cancel and five remain, so the exponents subtract in that order. Testing with x equal to two gives five hundred and twelve over sixteen, which is thirty-two, matching two to the fifth.
Check
Both parts of the fraction.
Check your understanding
Simplify the quantity 3 over y, squared.
Answer: A
Why: Both the numerator and the denominator receive the exponent, so three squared is nine and y squared is the denominator. Substituting y equal to three gives one, and only this answer produces one.
Check
The exponents subtract to zero.
Check your understanding
What is a^6 over a^6, for a not zero?
Answer: A
Why: Six minus six is zero, and a nonzero base to the zero power is one. Directly, any nonzero quantity divided by itself is one, so the two routes agree.
Real world
This is Exercise 59's situation. The average salary of a baseball player was about 4 times 10 to the fifth dollars in 1985 and about 6 times 10 to the fifth in 1990.
Discussion prompt
Compare the two salaries by their ratio, say what the answer means, and say what the corresponding difference would tell you instead. Then say which comparison would still make sense if the two figures were from very different decades.
Hint: Divide the coefficients and subtract the exponents.
Answer:
\[ \dfrac{6 \times 10^5}{4 \times 10^5} = 1.5 \times 10^0 = 1.5 \]
The 1990 salary was one and a half times the 1985 one, or fifty per cent higher. The difference is two hundred thousand dollars, which answers how much more rather than how many times.
The ratio is what survives a change of scale. Comparing a 1920s salary with a modern one, a difference in dollars would be dominated by inflation and by the sheer size of modern figures, while a ratio still says how many times larger — which is why growth is almost always reported proportionally.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
What is y^3 divided by y^5?
Correct: 1/y^2, since 3 minus 5 is -2.
\[ \dfrac{y^3}{y^5} = y^{-2} = \dfrac{1}{y^2} \]
\[ \text{at } y = 2: \; \tfrac{8}{32} = \tfrac{1}{4} \;\checkmark \]
Why: The numerator's exponent comes first in the subtraction, giving negative two, which Lesson 8.2 turns into a reciprocal. Cancelling confirms it independently: three y's on top cancel three of the five below, leaving two underneath. The first option subtracts in whichever order avoids a negative, which produces the reciprocal of the truth — and testing with y equal to two settles it, since eight over thirty-two is a quarter rather than four.
Explain it
They know the multiplication rules and are guessing at the division ones.
Discussion prompt
In no more than four sentences, explain what dividing powers does to the exponents and why. Then tell them what to do when the answer comes out with a negative exponent.
Hint: Cancelling factors.
Answer:
A usable answer: write the factors out and cancel — every factor on the bottom knocks out one on the top, so what is left is the difference between the two counts. That is why dividing subtracts the exponents, and it is the opposite of multiplying, which added them.
If the bottom has more factors than the top, the subtraction gives a negative number and the leftover factors are underneath. Write it as one over the base to the positive power, because a simplified answer is expected to have no negative exponents in it.
Exit ticket
Name the weakest spot before you close the deck. That is the one worth ten minutes tonight.
Predict first
Which of these would you least want to be handed cold on a quiz tomorrow?
Correct: Whichever you picked is the right answer — and each one has a specific fix.
Why: The order is fixed by remembering that the numerator's exponent comes first, and by cancelling factors when in doubt. Raising both parts is fixed by substituting a number for the variable and checking the fraction. Reciprocating is fixed by writing the negative exponent, then the reciprocal, as separate steps. Choosing the rule is fixed by naming the structure before touching the exponents. Pick yours and do five of that kind tonight rather than twenty mixed ones.
Connect it up
Do this on paper. It is worth more than rereading the slides.
Draw it
At the top of a page write four to the fifth over four cubed as individual factors, cross out the cancelling ones, and write the rule it demonstrates. Underneath, simplify three quotients: one giving a positive exponent, one giving zero and one giving a negative exponent, rewriting the last with a positive exponent and checking all three by cancelling. In the middle, expand a quotient raised to a power fully, showing that both the numerator and the denominator receive the exponent, and beside it simplify one expression that needs three of the chapter's four rules, naming each rule as you use it. In the lower half, write two quantities as coefficients times powers of ten and compare them by ratio, then by difference, and write one sentence saying what each comparison answers. Finally, in the margin, write the four rules as a table with a column for what happens to the exponents.
Every one of your cancelling checks should give the same answer the rule did. If one does not, the subtraction was done in the wrong order — which the cancelling makes visible immediately.
Recap
Five things, and the first is the one whose order matters.
| If the question says | Your first move is |
|---|---|
| Two powers with the same base, divided | Subtract the exponents, top minus bottom |
| The difference comes out negative | Rewrite as a reciprocal |
| A fraction inside brackets with an exponent | Raise the top and the bottom |
| The numerator is a sum | Split the fraction into separate quotients |
| How many times larger | Compute the ratio, not the difference |
Lesson 8.5 puts these rules to work. Scientific notation writes every number as a coefficient times a power of ten, so multiplying and dividing large or small numbers reduces to small arithmetic on the coefficients and on the exponents.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.4 Division Properties of Exponents §8.4, pp. 462-468 — everything on these slides traces back here
Want this taught 1-on-1? Alexander tutors Algebra 1 — $55/session, free consultation.