Unit 8 · Lesson 2 of 6
Rational exponents
You'll be able to: rewrite and evaluate powers with fraction exponents, like \(8^{2/3}\), and simplify them with the exponent rules you already know.
Step 1 of 5Warm-up
Two quick questions from earlier lessons. Answer from memory first. Pulling it out of your head is what makes it stick.
\(\sqrt[3]{-64}\)
Look for a number that, used three times, gives \(-64\). Since \(4 \cdot 4 \cdot 4 = 64\), try \(-4\): \((-4)(-4)(-4) = 16 \cdot (-4) = -64\). Answer: \(-4\).
Simplify \(x^{-4} \cdot x^{9}\).
Same base being multiplied, so add the exponents: \(-4 + 9 = 5\). Answer: \(x^5\).
Step 2 of 5The idea
A rational exponent is an exponent that is a fraction, like the \(\tfrac{1}{2}\) in \(9^{1/2}\). It is just another way to write a root.
The bottom of the fraction is the index of the root (which root). The top is an ordinary power.
Bottom = root. \(a^{1/n} = \sqrt[n]{a}\). For example, \(9^{1/2} = \sqrt{9} = 3\) and \(8^{1/3} = \sqrt[3]{8} = 2\).
Top = power. \(a^{m/n} = \left(\sqrt[n]{a}\right)^m\). Take the root first, then the power. The numbers stay small that way.
Negative = flip. \(a^{-m/n} = \dfrac{1}{a^{m/n}}\). The minus sign moves the power to the bottom of a fraction. It does not make the answer negative.
The old rules still work. \(x^a \cdot x^b = x^{a+b}\), \(\dfrac{x^a}{x^b} = x^{a-b}\), \((x^a)^b = x^{ab}\), even when a and b are fractions.
Why does \(\tfrac{1}{2}\) mean square root? Square it with the power rule: \(\left(9^{1/2}\right)^2 = 9^{\frac12 \cdot 2} = 9^1 = 9\). So \(9^{1/2}\) is the number whose square is 9, which is \(\sqrt{9}\).
A memory aid: "power over root". In \(a^{m/n}\), the top m is the power and the bottom n is the root.
Why this shows up on the placement test: you may be asked for a value like \(16^{3/4}\) or to match \(x^{2/3}\) with its radical form, and these exponents come back in Unit 11.
Step 3 of 5Worked examples
Example 1: \(27^{2/3}\), then \(16^{-3/4}\)
- Bottom 3 means cube root, top 2 means square: \(27^{2/3} = \left(\sqrt[3]{27}\right)^2\).
- \(\sqrt[3]{27} = 3\) (because \(3 \cdot 3 \cdot 3 = 27\)).
- \(3^2 = 9\). So \(27^{2/3} = 9\).
- For \(16^{-3/4}\), the minus sign means flip first: \(16^{-3/4} = \dfrac{1}{16^{3/4}}\).
- \(16^{3/4} = \left(\sqrt[4]{16}\right)^3 = 2^3 = 8\) (\(2 \cdot 2 \cdot 2 \cdot 2 = 16\)), so \(16^{-3/4} = \dfrac{1}{8}\).
Example 2: simplify \(x^{1/2} \cdot x^{1/3}\), then \(\left(8x^{6}\right)^{2/3}\) (x is positive)
- Same base being multiplied, so add the exponents: \(\tfrac12 + \tfrac13 = \tfrac36 + \tfrac26 = \tfrac56\).
- So \(x^{1/2} \cdot x^{1/3} = x^{5/6}\).
- For the second one, every factor inside gets the exponent: \(\left(8x^6\right)^{2/3} = 8^{2/3} \cdot \left(x^6\right)^{2/3}\).
- \(8^{2/3} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4\).
- \(\left(x^6\right)^{2/3} = x^{6 \cdot \frac23} = x^4\) (power of a power: multiply the exponents). Answer: \(4x^4\).
Step 4 of 5Practice
Type your answer and press Check. If you miss twice, open the worked solution. Then mark how it went. In every problem, assume all variables are positive.
\(25^{1/2}\)
Bottom 2 means square root: \(25^{1/2} = \sqrt{25} = 5\).
Write \(\sqrt[5]{x^{3}}\) using a rational exponent.
The index 5 goes on the bottom (root), the power 3 goes on top. Answer: \(x^{3/5}\).
\(8^{2/3}\)
Cube root first: \(\sqrt[3]{8} = 2\). Then square: \(2^2 = 4\). Answer: \(4\).
\(81^{-3/4}\)
Negative means flip: \(81^{-3/4} = \dfrac{1}{81^{3/4}}\).
\(\sqrt[4]{81} = 3\) (because \(3 \cdot 3 \cdot 3 \cdot 3 = 81\)), and \(3^3 = 27\). Answer: \(\dfrac{1}{27}\).
\(\left(\dfrac{4}{9}\right)^{3/2}\)
Square root of the top and the bottom: \(\sqrt{\dfrac{4}{9}} = \dfrac{2}{3}\).
Then cube: \(\left(\dfrac{2}{3}\right)^3 = \dfrac{8}{27}\). Answer: \(\dfrac{8}{27}\).
Simplify \(x^{2/3} \cdot x^{1/4}\). The answer has the form \(x^{n}\). What number is \(n\)?
Same base being multiplied, so add the exponents. Common denominator 12: \(\tfrac23 + \tfrac14 = \tfrac{8}{12} + \tfrac{3}{12} = \tfrac{11}{12}\).
Answer: \(x^{11/12}\), so \(n = \dfrac{11}{12}\).
Simplify \(\left(16x^{8}\right)^{3/4}\).
Give the exponent to each factor: \(16^{3/4} \cdot \left(x^{8}\right)^{3/4}\).
\(16^{3/4} = \left(\sqrt[4]{16}\right)^3 = 2^3 = 8\). And \(\left(x^8\right)^{3/4} = x^{8 \cdot \frac34} = x^6\).
Answer: \(8x^6\).
Which expression is equal to \(x^{-3/2}\)?
- \(\dfrac{1}{\sqrt[3]{x^{2}}}\)
- \(\dfrac{1}{\sqrt{x^{3}}}\)
- \(\dfrac{-1}{\sqrt{x^{3}}}\)
- \(\dfrac{-1}{\sqrt[3]{x^{2}}}\)
The minus sign means flip (it does not make the answer negative): \(x^{-3/2} = \dfrac{1}{x^{3/2}}\).
"Power over root": the top 3 is the power, the bottom 2 is a square root, so \(x^{3/2} = \sqrt{x^3}\).
Answer: (b) \(\dfrac{1}{\sqrt{x^{3}}}\). Choices (c) and (d) wrongly make it negative; (a) and (d) swap the root and the power.
Step 5 of 5Wrap-up
You can now read a fraction exponent as "power over root", turn a negative exponent into a fraction, and use the exponent rules with fractions.
Read or watch next: OpenStax Intermediate Algebra 2e, 8.3 Simplify Rational Exponents (free textbook, with more practice at the end of the section).
Also good: Khan Academy: Algebra 2, Rational exponents (short videos). Quick rules: Radicals and complex numbers cheat sheet.
Something unclear? Ask your teacher bot a follow-up question, for example: "Why is \(8^{-1/3}\) equal to one half and not negative two?" It knows this course and can explain things another way.