Rational exponents worksheet for Precalculus

Precalculus pushes rational exponents into negative powers and coefficients, moves between radical and exponent form, and solves equations by raising both sides to a reciprocal power. This is the notation power functions and their inverses are written in.

Also taught in Algebra 2 and Algebra 1, each at its own level and with its own examples.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name Rational Exponents Date Period Rewrite, evaluate, simplify, or solve each. When rewriting, assume every variable is positive. Convert between roots and rational exponents Write in simplest radical form: $r^{\frac{11}{6}}$ Write using a rational exponent: $\sqrt[7]{x^{8}}$ Write using a rational exponent: $\sqrt[6]{r^{11}}$ Write in simplest radical form: $a^{\frac{10}{7}}$ Write using a rational exponent: $\sqrt[7]{n^{9}}$ Evaluate rational exponents $2401^{- \frac{3}{4}}$ $16807^{- \frac{1}{5}}$ $256^{- \frac{1}{2}}$ $125^{- \frac{4}{3}}$ $343^{- \frac{5}{3}}$ Simplify with rational exponents $\frac{x^{\frac{5}{2}}}{x^{\frac{1}{3}}}$ ${(9 x^{2})}^{\frac{1}{2}}$ ${(8 x^{3})}^{\frac{2}{3}}$ $\frac{x^{\frac{4}{3}}}{x^{\frac{1}{3}}}$ $\frac{x^{\frac{1}{2}}}{x^{\frac{1}{4}}}$ Solve equations with rational exponents $3 {(x + 4)}^{\frac{2}{3}} + 3 = 78$ ${(x + 2)}^{\frac{3}{2}} = 125$ $2 {(x - 4)}^{\frac{3}{4}} = 16$ ${(x - 1)}^{\frac{4}{3}} + 3 = 19$ ${(x + 1)}^{\frac{2}{3}} - 5 = - 1$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Rational Exponents

Date Period

Rewrite, evaluate, simplify, or solve each. When rewriting, assume every variable is positive.

Convert between roots and rational exponents
Write in simplest radical form:
$r^{\frac{11}{6}}$
Write using a rational exponent: $\sqrt[7]{x^{8}}$
Write using a rational exponent: $\sqrt[6]{r^{11}}$
Write in simplest radical form:
$a^{\frac{10}{7}}$
Write using a rational exponent: $\sqrt[7]{n^{9}}$

Evaluate rational exponents
$2401^{- \frac{3}{4}}$
$16807^{- \frac{1}{5}}$
$256^{- \frac{1}{2}}$
$125^{- \frac{4}{3}}$
$343^{- \frac{5}{3}}$

Simplify with rational exponents
$\frac{x^{\frac{5}{2}}}{x^{\frac{1}{3}}}$
${(9 x^{2})}^{\frac{1}{2}}$
${(8 x^{3})}^{\frac{2}{3}}$
$\frac{x^{\frac{4}{3}}}{x^{\frac{1}{3}}}$
$\frac{x^{\frac{1}{2}}}{x^{\frac{1}{4}}}$

Solve equations with rational exponents
$3 {(x + 4)}^{\frac{2}{3}} + 3 = 78$
${(x + 2)}^{\frac{3}{2}} = 125$
$2 {(x - 4)}^{\frac{3}{4}} = 16$
${(x - 1)}^{\frac{4}{3}} + 3 = 19$
${(x + 1)}^{\frac{2}{3}} - 5 = - 1$



Sigma Prep · sigmaprep.io/worksheets Name Rational ExponentsAnswer keyVersion 2 Date Period Rewrite, evaluate, simplify, or solve each. When rewriting, assume every variable is positive. Convert between roots and rational exponents Write in simplest radical form: $r^{\frac{11}{6}}$ $r \sqrt[6]{r^{5}}$ Write using a rational exponent: $\sqrt[7]{x^{8}}$ $x^{\frac{8}{7}}$ Write using a rational exponent: $\sqrt[6]{r^{11}}$ $r^{\frac{11}{6}}$ Write in simplest radical form: $a^{\frac{10}{7}}$ $a \sqrt[7]{a^{3}}$ Write using a rational exponent: $\sqrt[7]{n^{9}}$ $n^{\frac{9}{7}}$ Evaluate rational exponents $2401^{- \frac{3}{4}}$ $\frac{1}{343}$ $16807^{- \frac{1}{5}}$ $\frac{1}{7}$ $256^{- \frac{1}{2}}$ $\frac{1}{16}$ $125^{- \frac{4}{3}}$ $\frac{1}{625}$ $343^{- \frac{5}{3}}$ $\frac{1}{16807}$ Simplify with rational exponents $\frac{x^{\frac{5}{2}}}{x^{\frac{1}{3}}}$ $x^{\frac{13}{6}}$ ${(9 x^{2})}^{\frac{1}{2}}$ $3 x$ ${(8 x^{3})}^{\frac{2}{3}}$ $4 x^{2}$ $\frac{x^{\frac{4}{3}}}{x^{\frac{1}{3}}}$ $x$ $\frac{x^{\frac{1}{2}}}{x^{\frac{1}{4}}}$ $x^{\frac{1}{4}}$ Solve equations with rational exponents $3 {(x + 4)}^{\frac{2}{3}} + 3 = 78$ ${- 129, 121}$ ${(x + 2)}^{\frac{3}{2}} = 125$ ${23}$ $2 {(x - 4)}^{\frac{3}{4}} = 16$ ${20}$ ${(x - 1)}^{\frac{4}{3}} + 3 = 19$ ${- 7, 9}$ ${(x + 1)}^{\frac{2}{3}} - 5 = - 1$ ${- 9, 7}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Rational ExponentsAnswer keyVersion 2

Date Period

Rewrite, evaluate, simplify, or solve each. When rewriting, assume every variable is positive.

Convert between roots and rational exponents
Write in simplest radical form:
$r^{\frac{11}{6}}$ $r \sqrt[6]{r^{5}}$
Write using a rational exponent: $\sqrt[7]{x^{8}}$ $x^{\frac{8}{7}}$
Write using a rational exponent: $\sqrt[6]{r^{11}}$ $r^{\frac{11}{6}}$
Write in simplest radical form:
$a^{\frac{10}{7}}$ $a \sqrt[7]{a^{3}}$
Write using a rational exponent: $\sqrt[7]{n^{9}}$ $n^{\frac{9}{7}}$

Evaluate rational exponents
$2401^{- \frac{3}{4}}$ $\frac{1}{343}$
$16807^{- \frac{1}{5}}$ $\frac{1}{7}$
$256^{- \frac{1}{2}}$ $\frac{1}{16}$
$125^{- \frac{4}{3}}$ $\frac{1}{625}$
$343^{- \frac{5}{3}}$ $\frac{1}{16807}$

Simplify with rational exponents
$\frac{x^{\frac{5}{2}}}{x^{\frac{1}{3}}}$ $x^{\frac{13}{6}}$
${(9 x^{2})}^{\frac{1}{2}}$ $3 x$
${(8 x^{3})}^{\frac{2}{3}}$ $4 x^{2}$
$\frac{x^{\frac{4}{3}}}{x^{\frac{1}{3}}}$ $x$
$\frac{x^{\frac{1}{2}}}{x^{\frac{1}{4}}}$ $x^{\frac{1}{4}}$

Solve equations with rational exponents
$3 {(x + 4)}^{\frac{2}{3}} + 3 = 78$ ${- 129, 121}$
${(x + 2)}^{\frac{3}{2}} = 125$ ${23}$
$2 {(x - 4)}^{\frac{3}{4}} = 16$ ${20}$
${(x - 1)}^{\frac{4}{3}} + 3 = 19$ ${- 7, 9}$
${(x + 1)}^{\frac{2}{3}} - 5 = - 1$ ${- 9, 7}$

How to do these

${(2 x + 1)}^{\frac{3}{2}} = 27$

Raise both sides to the 2/3 power: 2x + 1 = 27^(2/3).
Cube root of 27 is 3, and 3² = 9.
2x + 1 = 9, so x = 4.

The answer is ${4}$ .

Where students go wrong

Raising both sides to the same power instead of its reciprocal. To undo a 3/2 power you raise to 2/3; using 3/2 again makes the exponent 9/4, not 1.

Also called fractional exponents or roots as powers.

What you can put on this worksheet

Swap between a root and a fractional power: Write using a rational exponent: $\sqrt[3]{v^{4}}$ → $v^{\frac{4}{3}}$
Work out the value: $64^{\frac{4}{3}}$ → $256$
Simplify with rational exponents: $x^{\frac{3}{4}} \cdot x^{\frac{1}{3}}$ → $x^{\frac{13}{12}}$
Solve equations with rational exponents: $x^{\frac{1}{2}} = 3$ → ${9}$
Number bases, coefficients, negative and decimal exponents: Write in radical form:
${(7 v)}^{\frac{3}{2}}$ → ${(\sqrt{7 v})}^{3}$
Coefficients, negative and decimal powers, two letters: $4 v \cdot 2 v^{\frac{2}{3}} \cdot 3 v^{- 3}$ → $\frac{24}{v^{\frac{4}{3}}}$
Isolate the power, negative exponents, 1/2 powers: $- v^{\frac{1}{3}} = - 2$ → ${8}$

Questions about these worksheets

Yes. Every rational exponents sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the rational exponents questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 7: swap between a root and a fractional power, work out the value, simplify with rational exponents, solve equations with rational exponents, number bases, coefficients, negative and decimal exponents, coefficients, negative and decimal powers, two letters and isolate the power, negative exponents, 1/2 powers. Tick as many as you want and set how many of each, or let it spread them evenly.

Doing the power first on a big number. 8^(2/3) is 4, and getting there via 64 is harder than it needs to be. Root first, then power, every time.

Rational exponents worksheet for Precalculus

Select your difficulty

Convert between roots and rational exponents

Evaluate rational exponents

Simplify with rational exponents

Solve equations with rational exponents

Convert between roots and rational exponents

Evaluate rational exponents

Simplify with rational exponents

Solve equations with rational exponents

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these rational exponents worksheets free?+

Do the rational exponents worksheets come with an answer key?+

Are these rational exponents worksheets printable?+

What grade are these rational exponents worksheets for?+

Can I get a different version of the same rational exponents worksheet?+

Is this a free alternative to Kuta Software for rational exponents?+

Can I choose how hard the rational exponents questions are?+

What kinds of question are on a rational exponents worksheet?+

What do students get wrong with rational exponents?+