Chapter 8 of Algebra 1: Concepts and Skills, built for a visual learner. Every exponent rule rebuilt by drawing and counting factors, zero and negative exponents forced by a halving pattern, scientific notation as a travel log for the decimal point, and exponential growth and decay curves compared against straight lines.
Subject: Algebra 1 · 60 slides · symbolic lesson
Open the interactive version of this deck · Homework for this lesson
Title
Algebra 1 · Chapter 8
Rules that come from counting factors, plus the curve that outgrows every straight line
Objectives
Every rule in this chapter can be rebuilt by writing the factors out. None of it needs memorising if you can count.
Figure (svg): A steeply rising exponential curve beside a straight line, showing the exponential overtaking
Section
Section 8.1
Concept
When two powers of the same base are multiplied, all their factors pool together, so the counts add.
Figure (svg): Two powers of x being multiplied, with their factors laid out end to end and counted to give the sum of the exponents
\[ x^3 \cdot x^2 = x^{3+2} = x^5 \]
Worked example
Simplify the expression below.
\[ (2x^3)(5x^4) \]
Multiply the coefficients separately from the powers
Why: The numbers 2 and 5 are ordinary factors and multiply normally; only the x powers use the exponent rule.
\[ 10 \cdot x^3 \cdot x^4 \]
Add the exponents of the matching base
Why: Three factors of x and four more makes seven factors altogether.
\[ 10x^7 \]
Figure (svg): Coefficients multiplied on one track and exponents added on another, meeting at the answer
Verify: test with x equal to 2
Why: The original gives 2 times 8, which is 16, times 5 times 16, which is 80, and 16 times 80 is 1280. The answer gives 10 times 128, which is also 1280.
Concept
Raising a power to another power means repeating the whole thing, so the exponents multiply.
\[ (x^3)^2 = x^3 \cdot x^3 = x^6 \]
Figure (svg): A power of x cubed being squared, expanded into two groups of three factors making six
Discrimination
The two rules look alike and do opposite things. Sort each expression by which applies.
Sort into buckets
Which rule does each expression need?
Error analysis
Find what went wrong here.
Annotate
On: \( (3x^2)^3 \;\overset{?}{=}\; 3x^6 \)
An exponent outside a bracket reaches every factor inside it, coefficients included.
Fill the middle
Fill each blank.
Fill in the blanks
(4x^2)(3x^5) = 12x^7} \;\text20\; (x^4)^5 = x^___}
Why: In the first, the coefficients multiply to 12 and the exponents add to 7. In the second, a power raised to a power multiplies the exponents to give 20. Adding in the second case would give 9, which is the single most common exponent error in the chapter.
Explain it to yourself
Do not quote the rule. Explain it.
\[ x^3 \cdot x^2 = x^5 \]
Discussion prompt
Why does multiplying powers of the same base add the exponents rather than multiplying them?
Hint: Write both powers out as products and count the letters.
Answer:
Because the exponent counts factors. Three x's multiplied by two more x's is simply five x's in one long product, and counting them gives five.
Multiplying the exponents would give six, which would mean six factors — but you never created six. The rule is a shortcut for counting, and if you ever forget which rule applies, writing the factors out settles it in seconds.
Section
Section 8.2
Concept
A zero exponent gives one, and a negative exponent gives the reciprocal. Neither is a convention — the pattern of halving leaves no alternative.
Figure (svg): A staircase of powers of two descending past two to the first, showing the pattern forcing two to the zero to be one
\[ a^0 = 1 \qquad a^{-n} = \frac{1}{a^n} \]
Worked example
Write the expression below with only positive exponents.
\[ \frac{3x^{-2}}{y^{-4}} \]
Move each negatively-exponented factor across the fraction bar
Why: A negative exponent means the factor belongs on the other side of the bar, where it becomes positive. The coefficient 3 has no negative exponent and does not move.
\[ \frac{3y^4}{x^2} \]
Check that no negative exponents remain
Why: Both exponents are now positive, so the expression is in the required form.
Figure (svg): Factors with negative exponents crossing the fraction bar and turning positive
Verify: test with x equal to 2 and y equal to 1
Why: The original gives 3 times one quarter over 1, which is three quarters. The answer gives 3 times 1 over 4, which is also three quarters.
Prediction
Commit before reasoning.
\[ 7^0 \qquad (-5)^0 \qquad (2x)^0 \]
Predict first
What do these three expressions equal?
Correct: All three equal 1, provided the base is not zero.
Why: Following the halving pattern down, any base divided by itself once gives one, and that is exactly what a zero exponent describes. The base being negative or containing a variable makes no difference. The one exclusion is zero itself, because zero to the zero has no consistent value.
Pattern
One step per frame. Predict the next line before advancing.
Step through it
Each step divides by 3. What must 3 to the zero be, and what comes after?
Nobody chose these values. The pattern of dividing by the base at every step forces them.
Sorting
Sort each expression by its value. Do not compute exactly.
Sort into buckets
Which of these are greater than one, equal to one, or between zero and one?
Counterexample
A classmate says: a negative exponent makes the answer negative.
Discussion prompt
Find a counterexample and explain what the negative sign actually does.
Hint: Compute two to the negative third and look at the sign of the answer.
Answer:
\[ 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \]
One eighth is positive, so the claim is false. A negative exponent produces a reciprocal, not a negative number.
The confusion comes from the same symbol doing two jobs. A minus in front of the whole expression negates it; a minus in the exponent flips it. Those are entirely different operations that happen to share a symbol.
Section
Section 8.3
Concept
A linear function adds a fixed amount each step. An exponential function multiplies by a fixed amount, and that always wins in the long run.
Figure (svg): A steeply rising exponential curve beside a straight line, showing the exponential overtaking
\[ y = 2^x \]
Worked example
Make a table for the function below and describe its shape.
\[ y = 2^x \]
Substitute several inputs, including negatives
Why: Negative inputs are where the interesting behaviour is, so do not skip them.
| x | 2 to the x | y |
|---|---|---|
| -2 | 1 over 4 | 0.25 |
| -1 | 1 over 2 | 0.5 |
| 0 | 1 | 1 |
| 1 | 2 | 2 |
| 3 | 8 | 8 |
Plot and join with a smooth curve
Why: The points are not in a straight line, so the graph curves — and it curves upward more and more steeply.
Describe the left-hand behaviour
Why: As x goes further negative the outputs shrink towards zero but never reach it, so the curve hugs the horizontal axis without touching.
Figure (svg): The curve of two to the x, rising steeply on the right and flattening towards the axis on the left
Verify: check the value at x equal to 0
Why: Any base to the zero power is one, and the curve does pass through (0, 1). Every exponential of this form crosses the vertical axis at its starting value.
Tweak it
One number controls the whole shape of the curve.
Parameter explorer
Drag the base. What happens as it passes below 1, and what point never moves?
\[ y = {b}^x \]
Prediction
A linear function adds 10 each step. An exponential doubles, starting from 1.
Predict first
After how many steps does the doubling function overtake the adding one?
Correct: Around step 6 — and after that the gap grows enormously.
\[ 2^6 = 64 > 60 = 10 \cdot 6 \]
Why: At step 5 the linear function is at 50 and doubling has reached 32, so the line is still ahead. At step 6 doubling reaches 64 while the line is at 60, and from there the exponential pulls away without limit. Exponential growth always overtakes linear growth eventually, however slow its start.
Comparison
Fill the blanks. The difference is one word: add or multiply.
Comparison matrix
| linear | exponential | |
|---|---|---|
| what happens each step | add a fixed amount | multiply by a fixed amount |
| shape of the graph | a straight line | a curve that bends away |
| table test | differences between outputs are constant | ratios between outputs are constant |
The table test is the reliable one. Compute differences and ratios; whichever comes out constant tells you which kind of function you have.
Section
Section 8.4
Concept
When powers of the same base are divided, matching factors cancel above and below the line, so the counts subtract.
Figure (svg): A fraction of powers with matching factors cancelling one by one, leaving the difference of the exponents
\[ \frac{x^5}{x^2} = x^{5-2} = x^3 \]
Worked example
Simplify the expression below.
\[ \frac{12x^6y^2}{3x^2y^5} \]
Divide the coefficients
Why: Twelve over three is four, and this has nothing to do with the exponent rules.
Subtract the exponents for each base separately
Why: For x the count is six minus two, giving four. For y it is two minus five, giving negative three.
\[ 4x^4y^{-3} \]
Move the negative exponent below the bar
Why: A negative exponent means the factor belongs on the other side, where it becomes positive.
\[ \frac{4x^4}{y^3} \]
Figure (svg): Coefficients, x powers and y powers handled in three separate columns
Verify: test with x equal to 1 and y equal to 2
Why: The original gives 12 times 4 over 3 times 32, which is 48 over 96, or one half. The answer gives 4 over 8, which is also one half.
Error analysis
Find the error in this simplification.
Annotate
On: \( \frac{x^3}{x^7} \;\overset{?}{=}\; x^4 \)
Whenever the bottom has the larger exponent, the surviving factors stay below the line.
Matching
Subtract top minus bottom for each base.
Match the pairs
Why: The third one is worth noticing: identical powers cancel completely, and 5 minus 5 gives a zero exponent, which is one. That is another independent route to the zero-exponent rule, and it agrees with the halving pattern exactly.
Faded example
Fill each blank.
Fill in the blanks
\frac4-2 = 2a^4b^2}b^___} = \frac___}___
Why: The coefficients give 4, the a exponents give 7 minus 9 which is negative 2, and the b exponents give 3 minus 1 which is 2. The negative a exponent moves below the bar, leaving 4b squared over a squared. Each base is handled independently, which is the habit that makes these reliable.
Pattern
If you can count factors, you can rebuild every rule in this chapter.
Six rules, one idea. If you ever forget which applies, write four or five factors out and count them.
Explain it
A younger student refuses to believe that anything to the power zero is one.
Discussion prompt
Convince them in two sentences using the pattern rather than a rule.
Hint: What do you divide by each time the exponent drops by one?
Answer:
Something like: each time you drop the exponent by one you divide by the base, so going from 3 squared, which is 9, down to 3 to the first, which is 3, means dividing by 3 — and doing it once more gives 1.
Then point out the alternative: if 3 to the zero were 0, the pattern would have to jump from 3 straight to 0, which no amount of dividing by 3 will ever do.
Reverse engineer
Work backwards from the answer.
Fill in the blanks
x^7} \cdot x^5 = x^5 \;\text___\; (x^3)^___} = x^___
Why: The first is a product, so the exponents add and the missing one is 12 minus 5, which is 7. The second is a power of a power, so the exponents multiply and the missing one is 15 divided by 3, which is 5. Reading the rules backwards like this is a fast way to check you have the right one.
Edge cases
Push the exponential function towards its edge.
\[ y = 2^x \]
Discussion prompt
As x becomes more and more negative, what happens to y, and does the curve ever reach zero?
Hint: Can halving a positive number ever produce exactly zero?
Answer:
The outputs keep halving: one half, one quarter, one eighth, and so on. They shrink towards zero without limit.
But they never reach it. Halving a positive number always leaves a positive number, however many times you do it, so the curve gets arbitrarily close to the horizontal axis and never touches. That line is called an asymptote.
\[ 2^{-10} = \frac{1}{1024} \quad \text{small, but not zero} \]
Section
Section 8.5
Concept
Scientific notation writes a number as a single digit, a decimal part, and a power of ten. The exponent records how far the point moved.
Figure (svg): A large number written out in full and then in scientific notation, with the decimal point's journey marked
\[ 4{,}500{,}000 = 4.5 \times 10^6 \]
Worked example
Write 0.00037 in scientific notation, then convert 6.02 times ten to the 23 back to a plain number.
Place the point after the first non-zero digit
Why: That gives 3.7, which is the required form of the leading part.
Count how far the point travelled and in which direction
Why: It moved four places to the right, and moving right means a negative exponent because the original number was small.
\[ 0.00037 = 3.7 \times 10^{-4} \]
For the reverse, move the point by the exponent
Why: An exponent of 23 means moving the point 23 places to the right, giving a 24-digit whole number.
\[ 6.02 \times 10^{23} = 602{,}000{,}000{,}000{,}000{,}000{,}000{,}000 \]
Figure (svg): A number line of magnitudes from ten to the minus four up to ten to the sixth, with example quantities marked
Verify: check the direction of each move
Why: 0.00037 is smaller than one, so its exponent must be negative, and it is. The Avogadro number is enormous, so its exponent must be positive and large, and it is.
Prediction
Commit before converting.
\[ 2.9 \times 10^{-5} \]
Predict first
What is this as a plain decimal?
Correct: 0.000029
Why: A negative exponent means the number is small, so the point moves five places to the left of where it sits in 2.9. Counting carefully gives four zeros after the decimal point before the 2. Moving only four places is the usual slip, and it makes the answer ten times too big.
Sorting
Sort by size using the exponent alone.
Sort into buckets
Which of these are larger than one?
Estimation
Comparing sizes is what scientific notation is really for.
\[ 7.2 \times 10^{9} \quad \text{versus} \quad 3.1 \times 10^{11} \]
Predict first
Roughly how many times bigger is the second number?
Correct: About 40 times bigger.
\[ \frac{3.1 \times 10^{11}}{7.2 \times 10^{9}} \approx 43 \]
Why: The exponents differ by 2, which is a factor of 100, and the leading parts differ by roughly a factor of 0.43. Multiplying gives about 43. Reading only the leading digits would suggest the first is bigger, which is exactly the mistake scientific notation is designed to prevent.
Error analysis
A student converted a small number and got a large one. Diagnose it.
Annotate
On: \( 0.0052 \;\overset{?}{=}\; 5.2 \times 10^{3} \)
Big number, positive exponent. Small number, negative exponent. Check the sign before checking anything else.
Matching
Count the places, and check the sign against the size.
Match the pairs
Why: The third one is worth noting: a number already between one and ten needs no shift at all, so its exponent is zero. All four share the same leading digits, which shows how completely the exponent controls the size.
Section
Section 8.6
Concept
Exponential growth multiplies by the same factor every period. The model has a starting value and a growth factor bigger than one.
\[ y = a(1 + r)^t \]
Figure (svg): Two curves side by side: an exponential growth curve rising and a decay curve falling towards zero
Worked example
A population of 500 grows by 8 percent per year. Find the population after 10 years.
Identify the starting amount and the growth rate
Why: The starting value is 500 and the rate is 0.08, so the growth factor is 1.08.
Write the model
Why: The factor 1.08 is applied once per year, so it carries the exponent t.
\[ P = 500(1.08)^t \]
Substitute the number of years
Why: Ten years means the factor is applied ten times.
\[ P = 500(1.08)^{10} \approx 1079 \]
Figure (svg): A bar chart of a population growing by eight percent a year over ten years, with the bars accelerating
Verify: check the first year by hand
Why: Eight percent of 500 is 40, so after one year there should be 540, and the model gives 500 times 1.08, which is exactly 540.
Socratic
The rate is fixed at 8 percent, yet each year adds more than the last.
Discussion prompt
Why does a constant percentage produce a growing amount of increase?
Hint: Eight percent of what, in year one, and eight percent of what, in year ten?
Answer:
Because the percentage is taken of a larger number each time. Eight percent of 500 is 40, but eight percent of 1000 is 80.
That is the whole difference between exponential and linear growth. Linear adds the same amount; exponential adds the same proportion, and a fixed proportion of a growing quantity is itself growing.
Real world
You put 200 dollars in an account paying 5 percent interest per year, left untouched.
Discussion prompt
Write the model and say how it differs from earning a flat 10 dollars a year.
Hint: In year two, is the interest calculated on 200 or on 210?
Answer:
\[ A = 200(1.05)^t \]
A flat 10 dollars a year is linear: after 20 years you would have 400 dollars exactly.
Compounding gives 200 times 1.05 to the 20th, which is about 531 dollars. The difference comes entirely from earning interest on interest already earned — the same mechanism as the growing increment above.
Fill the middle
Turning a percentage into a multiplier is the step that decides everything.
Fill in the blanks
\text1.07 7\% \Rightarrow \text0.93 ___ \qquad \text___ 7\% \Rightarrow \text___ ___
Why: Growth keeps everything and adds more, so the factor is 1 plus the rate. Decay keeps what is left over, so the factor is 1 minus the rate. Using 0.07 as the decay factor is the classic error and would destroy 93 percent of the quantity in a single step rather than 7 percent.
Invariant
Watch a population grow and name the thing that never changes.
Step through it
The amount added changes every year. What does not?
The ratio is the invariant. That is the table test for an exponential: constant ratios, not constant differences.
Section
Section 8.7
Concept
Exponential decay works identically, except the factor is less than one so the quantity shrinks — approaching zero without ever reaching it.
\[ y = a(1 - r)^t \]
Figure (svg): Two curves side by side: an exponential growth curve rising and a decay curve falling towards zero
Worked example
A car worth 20000 dollars loses 15 percent of its value each year. What is it worth after 5 years?
Identify the starting value and the decay rate
Why: The starting value is 20000 and the rate is 0.15, so the decay factor is 1 minus 0.15, which is 0.85.
Write the model
Why: Each year the car keeps 85 percent of the previous year's value.
\[ V = 20000(0.85)^t \]
Substitute five years
Why: The factor is applied five times.
\[ V = 20000(0.85)^5 \approx 8874 \]
Figure (svg): A falling bar chart of a car's value over five years, each bar 85 percent of the previous one
Verify: check the first year against the percentage
Why: Fifteen percent of 20000 is 3000, so after one year the car should be worth 17000, and the model gives 20000 times 0.85, which is exactly 17000.
Prediction
The multiplier decides everything. Read it and commit.
Predict first
Which of these models describes decay?
Correct: y = 40 times 0.92 to the t.
Why: A multiplier below one shrinks the quantity each step, which is decay. Multipliers of 1.92 and 1.02 both grow, just at very different speeds. The last option is linear rather than exponential — the t is added, not used as an exponent.
Discrimination
Sort each situation by the kind of model it needs.
Sort into buckets
Linear, exponential growth, or exponential decay?
Trap
A 20000 dollar car loses 15 percent of its value each year. What is it worth after 5 years?
Compute 15 percent of the original value and subtract it five times
Why: Fifteen percent of 20000 is 3000, so five years of losses is 15000. This is how a flat depreciation would work.
\[ 20000 - 5(3000) = 5000 \]
This treats the loss as a fixed amount. But 15 percent of the second year's smaller value is less than 3000, so this answer is far too low.
Apply the percentage to each year's own value.
Multiply by 0.85 once per year
Why: Each year the car keeps 85 percent of whatever it was worth at the start of that year, not of its original price.
\[ 20000(0.85)^5 \approx 8874 \]
The difference is nearly 4000 dollars. Percentage change is always exponential, never linear, and this is the mistake that difference comes from.
Check
Solve it on paper before you click.
Check your understanding
Simplify (2x^3)^4 divided by (4x^5).
Answer: A
Why: The bracket raised to the 4th gives 2 to the 4th times x to the 12th, which is 16x to the 12th. Dividing by 4x to the 5th gives 16 over 4, which is 4, and 12 minus 5, which is 7. So the answer is 4x to the 7th.
Check
Solve it on paper before you click.
Check your understanding
A town of 8000 people shrinks by 4 percent per year. Which model is correct?
Answer: A
Why: Losing 4 percent means keeping 96 percent, so the decay factor is 0.96 applied once per year. After one year the model gives 7680, which is 8000 minus 4 percent of 8000.
Commit first
Answer, then rate your confidence.
\[ \left(\frac{x^4}{x^{-2}}\right)^{2} \]
Predict first
What does this simplify to?
Correct: x to the 12th.
\[ \left(x^{4-(-2)}\right)^2 = (x^6)^2 = x^{12} \]
Why: Inside the bracket the exponents subtract: 4 minus negative 2 is 6, giving x to the 6th. Raising that to the 2nd multiplies the exponents to give 12. The double negative in the subtraction is where this usually goes wrong — subtracting a negative adds.
Exit ticket
Last commitment of the chapter.
Predict first
Which of these is shakiest right now?
Correct: Whatever you picked is the one to drill first.
Why: The first two are pure rule-recognition and are fixed by writing factors out a dozen times. The fourth is different in kind — it needs you to convert a percentage into a multiplier, and getting 0.96 rather than 0.04 or 1.04 is the single most consequential step in any growth or decay problem.
Connect it up
One page, drawn by you.
Draw it
Write the idea an exponent counts factors in the middle of the page. Branch out to: multiplying, power of a power, dividing, zero exponent, negative exponent. On each branch write how counting factors produces that rule. Then add a separate region for exponential functions, with growth and decay, and connect it back to repeated multiplication.
If your map treats the rules as six separate facts rather than six consequences of one idea, redraw it — that is exactly the difference between memorising this chapter and understanding it.
Recap
You can now work with quantities that multiply rather than add, which is how most real growth actually behaves.
| if you remember one thing | it should be |
|---|---|
| about the rules | an exponent counts factors, so write them out when unsure |
| about negative exponents | they flip, they never negate |
| about brackets | an outside exponent reaches the coefficient too |
| about percentages | losing 15 percent means multiplying by 0.85, not by 0.15 |
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