Evaluating expressions with rational exponents. Includes cube roots and nth roots, why the nth root equals the one-over-n power, rational exponents of the form m over n and the two equivalent routes through them, the exponent properties applied to fractional exponents, and recovering a radius from a volume with a one-third power.
Subject: Algebra 1 · 65 slides · symbolic lesson
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Title
Algebra 1 · Chapter 12 — Radicals and More Connections to Geometry
Rational 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. 12 Radicals and More Connections to Geometry — Lesson 12.4 Rational Exponents §12.4, pp. 710-714 — the lesson these objectives are drawn from
Warm-up
Chapter 8 gave rules for exponents and Chapter 9 gave square roots. This lesson connects them by allowing an exponent to be a fraction.
Discussion prompt
What is two cubed? Now name a number whose cube is eight, and say how many such numbers there are.
Hint: Try a negative one as well.
Answer:
\[ 2^3 = 8, \quad \text{and } 2 \text{ is the only real number with } b^3 = 8 \]
Negative two cubed is negative eight, not eight, so cubing preserves the sign and there is exactly one real cube root. That is quite unlike a square root, which came in pairs, and it makes cube roots simpler to work with.
Concept
If b to the power n equals a, then b is an nth root of a. Writing that root as a to the power one over n makes the exponent rules of Chapter 8 apply to roots as well.
rational exponent — An exponent that is a fraction. For a non-negative a and positive integers m and n, a to the power m over n means the nth root of a, raised to the power m.
The value of a to the one over n is taken to be non-negative.
Figure (svg): Radical notation and rational exponent notation for the same thing
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.4 Rational Exponents §12.4, pp. 710-711
Section
Section 1
Concept
If b cubed equals a, then b is a cube root of a. More generally, if b to the power n equals a, then b is an nth root of a.
\[ b^3 = a \;\Longrightarrow\; b = \sqrt[3]{a} = a^{1/3} \]
Two is a cube root of eight because two cubed is eight.
Figure (svg): A cube root defined by undoing a cube
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.4 Rational Exponents §12.4, pp. 710-710 — the definitions of cube root and nth root, and Example 1
Picture it
One number each way.
Figure (svg): A cube root defined by undoing a cube
Cubing a negative gives a negative, so nothing is lost and the return trip has a single destination. Squaring destroyed the sign, which is why square roots needed a plus-or-minus and cube roots do not.
Worked example
This is Example 1 from the textbook.
\[ \text{Find } 27^{1/3}, \; \sqrt[3]{1000} \text{ and } 64^{1/2}. \]
Take the first
Why: Three cubed is twenty-seven.
\[ 3 \]
Take the second
Why: Ten cubed is a thousand.
\[ 10 \]
Take the third
Why: Eight squared is sixty-four.
\[ 8 \]
Note the sign convention
Why: The non-negative root is taken.
\[ 8, \text{ not } -8 \]
Figure (svg): Radical notation and rational exponent notation for the same thing
\[ 27^{1/3} = 3, \; \sqrt[3]{1000} = 10, \; 64^{1/2} = 8 \]
Verify: check by raising to the power
Why: Three cubed is twenty-seven, ten cubed is a thousand and eight squared is sixty-four. Raising the answer to the matching power is the natural check for any root.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.4 Rational Exponents §12.4, pp. 710-710
Matching
Check by raising to the power.
Match the pairs
Why: The last two both involve sixty-four and give different answers, since one asks for a square root and the other for a cube root. The index is what distinguishes them.
Worked example
Guided Practice 1 to 4.
\[ \text{Evaluate } \sqrt[3]{64}, \; 625^{1/2}, \; 225^{1/2} \text{ and } 216^{1/3}. \]
Take the cube root of sixty-four
Why: Four cubed is sixty-four.
\[ 4 \]
Take the square root of 625
Why: Twenty-five squared.
\[ 25 \]
Take the square root of 225
Why: Fifteen squared.
\[ 15 \]
Take the cube root of 216
Why: Six cubed.
\[ 6 \]
Figure (svg): Radical notation and rational exponent notation for the same thing
\[ 4, \; 25, \; 15, \; 6 \]
Verify: notice sixty-four appears twice
Why: The cube root of sixty-four is four and its square root is eight, so the same number has different roots depending on which is asked for. Reading the index carefully is the first thing to do.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.4 Rational Exponents §12.4, pp. 711-711
Trap
\[ \sqrt[3]{64} = 8 \]
Take the root of sixty-four
Why: Eight is the root that came to mind.
Eight is the square root, since eight squared is sixty-four. The cube root is four, because four cubed is sixty-four, and the two are quite different numbers.
\[ \sqrt[3]{64} = 4 \quad \text{since} \quad 4^3 = 64 \]
Read which root is being asked for, then check by raising to that power
Why: The index or the denominator says which.
Sixty-four is unusual in having both a tidy square root and a tidy cube root, which is why it is a favourite in exercises.
Faded example
The natural test for a root.
Fill in the blanks
216^3 = 6 \text216 6^___} = ___
Why: A root is checked by raising the answer to the matching power and recovering the original. That works for any index and needs no memorisation.
Prediction
Compare with square roots.
Predict first
How many real cube roots does a positive number have?
Correct: Exactly one.
\[ 2^3 = 8, \quad (-2)^3 = -8 \]
Why: Cubing preserves sign — a positive number cubed is positive and a negative one cubed is negative — so only one real number cubes to any given value. Squaring destroys sign, which is why a positive number has two square roots and needed a plus-or-minus. Cube roots are therefore simpler, and every real number including the negatives has exactly one.
Socratic
Square roots came in pairs.
Discussion prompt
Explain why cubing does not lose information the way squaring does. Then say what that means for negative numbers.
Hint: What is the sign of a cube?
Answer:
Squaring a number and squaring its negative give the same result, so the operation cannot be undone uniquely. Cubing does not: a positive number cubed stays positive and a negative one stays negative, so no two different numbers share a cube and the return trip is unambiguous.
It also means negative numbers have real cube roots, unlike square roots. The cube root of negative eight is negative two, whereas the square root of negative eight does not exist over the reals. Odd powers preserve sign and even powers destroy it, which is the whole difference.
Section
Section 2
Concept
Three copies of a to the one third multiply to a to the power one, which is a. That is exactly what a cube root must do, so the one-third power is the cube root.
\[ a^{1/3} \cdot a^{1/3} \cdot a^{1/3} = a^{1} = a \]
The definition is chosen so the product rule keeps working.
Figure (svg): Why the cube root is the one-third power
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.4 Rational Exponents §12.4, pp. 710-710 — the derivation of the rational exponent notation from the product of powers property
Picture it
The product rule decides.
Figure (svg): Why the cube root is the one-third power
Nothing was assumed here beyond Lesson 8.1's rule about adding exponents. Extending that rule to fractions determines what a fractional exponent must mean.
Worked example
The textbook's own justification.
\[ \text{Show that } a^{1/3} \text{ must be the cube root of } a. \]
Multiply three copies
Why: Using the product rule.
\[ a^{1/3} \cdot a^{1/3} \cdot a^{1/3} \]
Add the exponents
Why: Three thirds.
\[ a ^{\frac{1}{3} + \frac{1}{3} + \frac{1}{3}} \]
Simplify
Why: One.
\[ a ^{1} = a \]
Compare with the cube root
Why: Three copies multiplying to a.
Figure (svg): Why the cube root is the one-third power
\[ a^{1/3} = \sqrt[3]{a} \]
Verify: test with a number
Why: Eight to the one third should be two, and two times two times two is eight. The general argument and the specific case agree, as they must.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.4 Rational Exponents §12.4, pp. 710-710
Faded example
Three thirds make one.
Fill in the blanks
a^1/3 + 1/3 + 1/3 \cdot a^1 \cdot a^___ = a^___} = a^___} = a
Why: The product rule from Lesson 8.1 adds exponents whatever they are, and three thirds sum to one. That forces the one-third power to be the cube root.
Worked example
Extending beyond cube roots.
\[ \text{Why is } a^{1/n} \text{ the } n \text{th root of } a? \]
Multiply n copies
Why: Each with exponent one over n.
\[ a^{1/n} \text{, } n \text{ times} \]
Add the exponents
Why: n copies of one over n.
\[ a ^{\frac{n}{n}} \]
Simplify
Why: n over n is one.
\[ a ^{1} = a \]
Conclude
Why: That is what an nth root does.
\[ a^{1/n} = \sqrt[n]{a} \]
Figure (svg): Why the cube root is the one-third power
\[ a^{1/n} = \sqrt[n]{a} \]
Verify: test at a fourth root
Why: Sixteen to the one quarter should be two, and two to the fourth is sixteen. The argument works for any index because it only ever used the product rule.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.4 Rational Exponents §12.4, pp. 710-710
Error analysis
The student interpreted a fractional exponent.
Annotate
On: \( \begin{aligned} 27^{1/3} &= 27 \div 3 \\ &= 9 \end{aligned} \)
The reliable defence is to check by raising to the matching power: nine cubed should give back twenty-seven if nine were right, and it does not. Every root answer can be tested that way in a single multiplication.
Elimination
Reading fractional exponents.
Eliminate the wrong options
What is 27 to the power one third?
Survives elimination: A
Why: The denominator of the exponent gives the index of the root. Checking by cubing the answer settles it: three cubed is twenty-seven and nine cubed is far larger.
Hypothesis
A definition could be anything.
Predict first
What determines what a fractional exponent must mean?
Correct: The requirement that the exponent rules keep working.
\[ a^{1/3} \cdot a^{1/3} \cdot a^{1/3} = a^{1} = a \]
Why: If a to the one third is to obey the product rule, then three copies of it must give a to the power one, which is a — and that is precisely the defining property of a cube root. So the definition is forced rather than chosen, which is why it is worth deriving once instead of memorising. Definitions in mathematics are often selected to preserve rules that already work, and this is a clear example.
Socratic
Radical notation already existed.
Discussion prompt
Say what writing a root as a fractional exponent makes possible. Then say when the radical notation is still preferable.
Hint: Think about combining several roots.
Answer:
It brings roots inside the exponent rules, so a product of roots, a root of a root, or a root raised to a power can all be handled by adding or multiplying exponents rather than by separate rules. The next section shows this working, and without it each of those cases would need its own rule.
Radical notation remains clearer to read, especially when the index matters at a glance, and it is the conventional way to present a final answer in this course. So the usual approach is to convert to exponents for the working and back to radicals for the answer, which is why both notations are worth being fluent in.
Section
Section 3
Concept
For a non-negative a, a to the power m over n means the nth root of a, raised to the power m. The two operations may be done in either order.
\[ a^{m/n} = (a^{1/n})^m = (\sqrt[n]{a})^m \]
Taking the root first keeps the numbers small.
Figure (svg): Two equivalent routes through a rational exponent
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.4 Rational Exponents §12.4, pp. 711-711 — the Rational Exponents definition and Example 2
Picture it
Root first, or power first.
Figure (svg): Two equivalent routes through a rational exponent
Both routes are valid because the power of a power rule lets the exponents be applied in either sequence. By hand, rooting first is almost always the easier one.
Worked example
This is Example 2 from the textbook.
\[ \text{Evaluate } 16^{3/2} \text{ and } 8^{4/3}. \]
Split the first exponent
Why: Root two, then power three.
\[ (16 ^{\frac{1}{2}}) ^{3} \]
Evaluate it
Why: Four cubed.
\[ 64 \]
Split the second
Why: Cube root, then fourth power.
\[ (8 ^{\frac{1}{3}}) ^{4} \]
Evaluate it
Why: Two to the fourth.
\[ 16 \]
Figure (svg): Two equivalent routes through a rational exponent
\[ 16^{3/2} = 64, \qquad 8^{4/3} = 16 \]
Verify: try the other order on the first
Why: Sixteen cubed is four thousand and ninety-six, whose square root is sixty-four — the same answer by a much longer route. Rooting first kept every number below a hundred.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.4 Rational Exponents §12.4, pp. 711-711
Faded example
Split the exponent.
Fill in the blanks
8^3 = (8^4}})^___} = 2^4 = 16
Why: The denominator three means a cube root and the numerator four means a fourth power. Taking the cube root first gives two, which is much easier to raise than eight would be.
Worked example
Guided Practice 5, 6 and 8.
\[ \text{Evaluate } 64^{3/2}, \; (\sqrt[3]{27})^2 \text{ and } 1000^{2/3}. \]
Take the first
Why: Square root, then cube.
\[ 8 ^{3} = 512 \]
Take the second
Why: Cube root of twenty-seven, squared.
\[ 3 ^{2} = 9 \]
Take the third
Why: Cube root of a thousand, squared.
\[ 10 ^{2} = 100 \]
Note the pattern
Why: Root by the denominator each time.
Figure (svg): Two equivalent routes through a rational exponent
\[ 512, \quad 9, \quad 100 \]
Verify: check the first the other way
Why: Sixty-four cubed is two hundred and sixty-two thousand one hundred and forty-four, whose square root is five hundred and twelve. The answer agrees, and the arithmetic was far heavier.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.4 Rational Exponents §12.4, pp. 711-711
Trap
\[ 16^{3/2} = (16^{1/3})^2 \]
Take the cube root and square it
Why: The three and the two were both present, so their roles were guessed.
The denominator gives the index of the root, so a two on the bottom means a square root. This version gives about 6.35, whereas the correct answer is sixty-four.
\[ 16^{3/2} = (16^{1/2})^3 = 4^3 = 64 \]
Root by the denominator, then raise to the numerator
Why: Bottom for the root, top for the power.
The one-over-n form makes this unambiguous: the n is always on the bottom.
Sorting
Read the denominator.
Sort into buckets
Sort each expression by the root it requires.
Only the denominator was consulted, never the numerator. The numerator decides what power to raise the root to, which is a separate question.
Prediction
Root first or power first.
Predict first
For 64 to the power 3/2, which route keeps the arithmetic smaller?
Correct: Take the square root first, then cube.
\[ (64^{1/2})^3 = 8^3 = 512 \qquad (64^3)^{1/2} = 262\,144^{1/2} = 512 \]
Why: Rooting first gives eight, and eight cubed is five hundred and twelve — three easy steps. Cubing first gives two hundred and sixty-two thousand and more, whose square root then has to be found. Both orders are valid and give the same answer, but by hand the difference in effort is considerable, and it grows quickly as the numbers get larger.
Socratic
Two operations, applied in sequence.
Discussion prompt
Explain why rooting then powering gives the same result as powering then rooting. Then say why one order is still preferred.
Hint: Which exponent rule applies?
Answer:
Both routes amount to applying the exponent m over n, and the power of a power rule says the exponents multiply — so one third times four is the same as four times one third. Multiplication can be done in either order, so the two routes are genuinely identical.
The preference is purely practical: taking the root first keeps every intermediate number small, whereas powering first can produce numbers with many digits whose root must then be found. For eight to the four thirds the difference is between working with two and working with four thousand and ninety-six.
Section
Section 4
Concept
The product of powers, power of a power and power of a product rules all hold for rational exponents, provided the bases are non-negative.
The rules never required whole-number exponents.
Figure (svg): The exponent properties applied to rational exponents
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.4 Rational Exponents §12.4, pp. 711-712 — the Properties of Rational Exponents summary and Example 3
Picture it
Fractions in place of integers.
Figure (svg): The exponent properties applied to rational exponents
Extending to fractions costs nothing because the rules were never about whole numbers in the first place. That is the whole reason the notation was worth introducing.
Worked example
This is Example 3 from the textbook.
\[ \text{Evaluate } 5^{1/3} \cdot 5^{2/3}, \; (7^{1/3})^6 \text{ and } (4 \cdot 25)^{1/2}. \]
Use the product rule
Why: One third plus two thirds.
\[ 5 ^{1} = 5 \]
Use the power of a power rule
Why: One third times six.
\[ 7 ^{2} = 49 \]
Use the power of a product rule
Why: Split across the factors.
\[ 4^{1/2} \cdot 25^{1/2} \]
Evaluate
Why: Two times five.
\[ 10 \]
Figure (svg): The exponent properties applied to rational exponents
\[ 5, \quad 49, \quad 10 \]
Verify: check the third directly
Why: Four times twenty-five is a hundred, whose square root is ten. Splitting the product first gave the same answer with smaller numbers, which is the point of the rule.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.4 Rational Exponents §12.4, pp. 712-712
Faded example
The operation on the bases decides.
Fill in the blanks
5^1 \cdot 5^2 = 5^___} = 5, \qquad (7^___)^6 = 7^___} = 49
Why: Multiplying the powers adds the exponents and raising a power to a power multiplies them. Which rule applies is decided by what is being done to the bases, not by the exponents themselves.
Worked example
When fractional exponents add to one.
\[ \text{Evaluate } 3^{1/2} \cdot 3^{3/2}. \]
Add the exponents
Why: A half plus three halves.
\[ 3 ^{2} \]
Evaluate
Why: Three squared.
\[ 9 \]
Check the other way
Why: Root three times three root three.
\[ 3\sqrt{3} \cdot \sqrt{3} \]
Compare
Why: Three times three.
\[ 9 \]
Figure (svg): The exponent properties applied to rational exponents
\[ 3^{1/2} \cdot 3^{3/2} = 3^2 = 9 \]
Verify: compare the two routes
Why: In radical form the product is root three times three root three, which is three times three, or nine. The exponent route needed one addition and the radical route needed simplifying first, which is the practical advantage of the notation.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.4 Rational Exponents §12.4, pp. 711-711
Trap
\[ 5^{1/3} \cdot 5^{2/3} = 5^{2/9} \]
Multiply the two exponents
Why: Two operations were present, so the exponents were combined by multiplying.
Multiplying powers of the same base adds the exponents; multiplying the exponents is what happens when a power is raised to a power. One third plus two thirds is one, so the answer is five.
\[ 5^{1/3} \cdot 5^{2/3} = 5^{1} = 5 \]
Add the exponents when the bases are multiplied
Why: Chapter 8's rule, unchanged.
The same confusion arises with whole-number exponents and has the same fix.
Matching
Three rules, three shapes.
Match the pairs
Why: Two of these use the same rule with different fractions. Recognising the shape rather than the numbers is what tells you which property applies.
Elimination
Using the exponent properties.
Eliminate the wrong options
What is (7 to the one third) to the sixth?
Survives elimination: A
Why: One third times six is two, so the result is seven squared. Each wrong option combines the numbers by a different operation, and only multiplication of the exponents is correct here.
Socratic
They were stated for whole numbers.
Discussion prompt
Explain why the exponent properties extend to fractional exponents without modification. Then say what condition on the base is needed.
Hint: What did the original proofs depend on?
Answer:
The rules describe how exponents combine, and the definition of a fractional exponent was chosen precisely so that they continue to hold — that is where a to the one over n came from in the first place. Nothing in the rules ever depended on the exponents being whole; extending the definition and preserving the rules were the same act.
The base must be non-negative, because a fractional exponent involves a root and even roots of negatives do not exist over the reals. That is why the properties are stated for non-negative a and b, and it is the one restriction that the whole-number case did not need.
Section
Section 5
Concept
A volume formula contains a cube of a length, so recovering the length from a volume means taking a cube root — which a one-third exponent performs directly.
\[ V = \tfrac{4}{3}\pi r^3 \;\Longrightarrow\; r = \left(\tfrac{3V}{4\pi}\right)^{1/3} \]
Rearranging first, then taking the root.
Figure (svg): The radius of a sphere recovered from its volume
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.4 Rational Exponents §12.4, pp. 710-714 — the lesson opener on finding the size of a shot from its volume
Picture it
Undoing a cube.
Figure (svg): The radius of a sphere recovered from its volume
The rearranging is ordinary Chapter 3 work and only the last step is new. A one-third power is what turns a volume back into a length.
Worked example
The lesson opener's situation, with a volume supplied.
\[ \text{A sphere has volume } 36\pi. \text{ Find its radius.} \]
Write the formula
Why: Volume of a sphere.
\[ V = \tfrac{4}{3}\pi r^3 \]
Substitute and rearrange
Why: Multiply by three over four pi.
\[ r^3 = \tfrac{3(36\pi)}{4\pi} \]
Simplify
Why: The pi cancels.
\[ r ^{3} = 27 \]
Take the cube root
Why: A one-third power.
\[ r = 3 \]
Figure (svg): The radius of a sphere recovered from its volume
\[ r = 27^{1/3} = 3 \]
Verify: substitute back
Why: Four thirds of pi times three cubed is four thirds of pi times twenty-seven, which is thirty-six pi. The answer reproduces the volume, which is the natural check for any rearranged formula.
Faded example
A one-third power.
Fill in the blanks
r^3 = 27 \;\Longrightarrow\; r = 27^3}} = 3
Why: The exponent's denominator matches the power being undone, so a cube needs a one-third power. Raising the answer back to the third power recovers twenty-seven.
Worked example
Reading the cube relationship.
\[ \text{If a sphere's radius doubles, what happens to its volume?} \]
Note the power
Why: Volume depends on r cubed.
\[ V \propto r^3 \]
Double the radius
Why: Two cubed.
\[ \times 8 \]
Reverse the question
Why: To double the volume.
\[ r \times 2^{1/3} \]
Evaluate
Why: About 1.26.
\[ \approx 26\% \text{ larger} \]
Figure (svg): The radius of a sphere recovered from its volume
\[ r \times 2 \;\Longrightarrow\; V \times 8 \]
Verify: check the reverse direction
Why: Doubling the volume needs the radius multiplied by the cube root of two, about 1.26 — only a twenty-six per cent increase in radius. Cubes grow fast, so their roots grow slowly, which is the same flattening seen with square roots but more pronounced.
Trap
\[ r^3 = 27 \;\Longrightarrow\; r = \sqrt{27} \approx 5.20 \]
Take a square root, as in earlier chapters
Why: Roots have mostly been square roots so far.
The variable is cubed, so a cube root undoes it. Checking settles it: four thirds of pi times 5.20 cubed is about a hundred and eighty-eight pi, not thirty-six pi.
\[ r = 27^{1/3} = 3 \]
Match the root to the power being undone
Why: A cube needs a cube root.
Substituting the answer back into the original formula catches the mismatch immediately.
Prediction
For a sphere.
Predict first
What happens to the volume?
Correct: It becomes eight times as large.
\[ 2^3 = 8, \qquad 2^{1/3} \approx 1.26 \]
Why: Volume depends on the cube of the radius, so doubling the radius multiplies the volume by two cubed, which is eight. That is why a small increase in the size of a ball makes a large difference to how much it holds, and conversely why doubling a volume requires the radius to grow by only about twenty-six per cent — the cube root of two.
Sorting
Match the index to the exponent.
Sort into buckets
Sort each equation by the root needed to solve it.
Volumes involve cubes and areas involve squares, which is why the two kinds of formula need different roots. Matching the index to the power is the whole decision.
Socratic
The formula could be rearranged another way.
Discussion prompt
Explain why recovering a radius from a volume needs a cube root. Then say what the same reasoning gives for an area.
Hint: What power does the formula apply to the radius?
Answer:
The volume formula raises the radius to the third power, so undoing it requires the inverse of cubing, which is the cube root. Written as an exponent, that is the one-third power, and multiplying the exponents three and one third gives one — the radius itself.
An area formula raises the radius to the second power, so recovering a radius from an area needs a square root, or a one-half power. The pattern is general: whatever power the formula applies, its reciprocal as an exponent undoes it, which is the whole content of the notation.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Square root | Cube root | |
|---|---|---|
| Written as | a to the one half | a to the one third |
| How many real roots | two for a positive number, none for a negative | exactly one, for any number |
| Undoes | squaring | cubing |
The middle row is the real difference. Even powers destroy sign and odd powers preserve it, which is why cube roots are unambiguous and square roots are not.
Pattern
To evaluate any expression with a rational exponent, these five moves cover it.
Step three is a preference rather than a requirement, but the difference between rooting sixteen and rooting four thousand and ninety-six is worth caring about.
OpenStax Elementary Algebra 2e, §9.8 Rational Exponents §9.8
Check
Match the root to the index.
Check your understanding
What is the cube root of 64?
Answer: A
Why: Four cubed is sixty-four, so four is the cube root. Eight is the square root, since eight squared is sixty-four.
Check
Bottom for the root, top for the power.
Check your understanding
Evaluate 8 to the power 4/3.
Answer: A
Why: The denominator three gives a cube root of eight, which is two, and the numerator four raises it to the fourth power, giving sixteen.
Check
Which rule applies?
Check your understanding
Evaluate 5 to the one third, times 5 to the two thirds.
Answer: A
Why: Multiplying powers of the same base adds the exponents, and one third plus two thirds is one, so the result is five.
Real world
This is the shot put question from the lesson opener. The shot is a metal sphere, and its size can be found from its volume using the formula for the volume of a sphere.
Discussion prompt
A sphere has volume 36 pi cubic units. Find its radius, and then say by how much the radius must grow to double the volume.
Hint: Rearrange, then take a cube root.
Answer:
\[ 36\pi = \tfrac{4}{3}\pi r^3 \;\Longrightarrow\; r^3 = 27 \;\Longrightarrow\; r = 27^{1/3} = 3 \]
Checking confirms it: four thirds of pi times three cubed is four thirds of pi times twenty-seven, which is thirty-six pi.
To double the volume the radius must be multiplied by the cube root of two, which is about 1.26 — a rise of only about twenty-six per cent. That is why a shot that looks only slightly larger can weigh considerably more, and it is the practical consequence of volume depending on the cube of a length while the length depends on only the cube root of the volume.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
Evaluate 16 to the power 3/2.
Correct: 64.
\[ 16^{3/2} = (16^{1/2})^3 = 4^3 = 64 \]
Why: The denominator two calls for a square root, giving four, and the numerator three raises that to the third power, giving sixty-four. The first option treats the exponent as a multiplier, which is the commonest misreading of fractional exponents and is checked instantly by raising the answer back — twenty-four to the two thirds is nowhere near sixteen. The third option swaps the numerator and denominator, taking a cube root when the two on the bottom calls for a square root. The last raises sixteen to the third without taking any root at all, which would be an exponent of three rather than three halves. Doing the root first is also what keeps the arithmetic manageable: four cubed is far easier than the square root of four thousand and ninety-six, though both give sixty-four.
Explain it
They said that twenty-seven to the one third is nine.
Discussion prompt
In no more than four sentences, explain what a fractional exponent means. Then give them a check they can run in one step.
Hint: It is a root, not a division.
Answer:
A usable answer: an exponent of one third means the cube root, not one third of the number. Twenty-seven to the one third asks for the number whose cube is twenty-seven, which is three — and it works out that way because three copies of a one-third power multiply to give the number itself.
The check is to raise your answer to the matching power. Three cubed is twenty-seven, which confirms it, whereas nine cubed is seven hundred and twenty-nine, which plainly does not.
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: Reading the exponent is fixed by remembering that the denominator gives the root and the numerator the power. The order is fixed by always rooting first. The properties are fixed by asking what is being done to the bases, since that decides whether exponents add or multiply. Recovering a length is fixed by matching the root's index to the power in the formula. 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 the definition of a cube root and beside it the derivation showing why the one-third power must mean it, with the exponents added explicitly. Underneath, evaluate four roots of your own, checking each by raising the answer to the matching power, and include one number that has both a tidy square root and a tidy cube root. In the middle, evaluate a power with a rational exponent by both available routes, writing the two columns side by side and ringing the largest number that appeared in each. Beneath that, write the three exponent properties with a fractional-exponent example for each, and beside them note what is being done to the bases in each case, since that is what decides whether exponents add or multiply. In the lower half, rearrange the volume formula for a sphere to give the radius, use it on a volume of your choosing, and write what happens to the volume when the radius doubles. Finally, in the margin, write the one condition on the base that fractional exponents require and did not before.
Every root on your page should be checked by raising it back to the matching power. That single habit catches both the divide-instead-of-root error and the wrong-index error, which are the two failures this lesson produces.
Recap
Five things, and the second explains why the first is written that way.
| If the question says | Your first move is |
|---|---|
| An exponent of 1/n | Take the nth root |
| An exponent of m/n | Root by n, then raise to m |
| Powers of the same base multiply | Add the exponents |
| A power is raised to a power | Multiply the exponents |
| A formula cubes a length | Undo it with a one-third power |
Lesson 12.5 returns to quadratics with a new technique. Completing the square deliberately builds a perfect square trinomial so that an equation can be solved by taking roots — and it is how the quadratic formula itself is derived.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.4 Rational Exponents §12.4, pp. 710-714 — everything on these slides traces back here
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