Free Printable Rational Exponents and Radicals

1.

Write with radical. Assume that all variables represent positive real numbers.

$\left(x^4y^4\right)^{\frac{1}{7}}$

a)

$\left(\sqrt[7]{xy}\right)^4$

b)

$x^{28}y^{28}$

c)

$\frac{1}{\left(\sqrt[7]{xy}\right)^4}$

d)

$\left(\sqrt[4]{xy}\right)^7$

2.

Write with radical. Assume that all variables represent positive real numbers.

$\left(5py^2\right)^{\frac{1}{3}}$

a)

$\sqrt[3]{5py}^{ }$

b)

$\sqrt[3]{5py^2}$

c)

$\sqrt[]{5py^2}$

d)

$\sqrt[5]{5py}^{ }$

3.

Write with radical. Assume that all variables represent positive real numbers.

$\left(7m+n\right)^{\frac{8}{9}}$

a)

$\sqrt[8]{\left(7m+n\right)^9}$

b)

$\sqrt[9]{\left(7m+n\right)^8}$

c)

$\sqrt[8]{7m+n^9}$

d)

$\sqrt[8]{7m^9+n^9}$

4.

Simplify by first converting to rational exponents. Assume that all variables represent positive real numbers.

$\sqrt[4]{s^7}.\sqrt[3]{s^5}$

a)

$s^{\frac{41}{12}}$

b)

$\sqrt[7]{s^{40}}$

c)

$s^{6.325}$

d)

$\sqrt[]{s^{35}}$

5.

A rectangular coal bin has a length of $\sqrt[]{45}$ feet and a width of $\sqrt[]{80}$ feet. What is its perimeter?

a)

$7\sqrt[]{5}\ feet$

b)

$14\sqrt[]{5}\ feet$

c)

$7\sqrt[]{4}\ feet$

d)

$14\sqrt[]{4}\ feet$

6.

Simplify by first converting to rational exponents. Assume that all variables represent positive real numbers.

$\sqrt[3]{64x^6y^3}$

a)

$3xy$

b)

$4x^2y$

c)

$2x^{18}y^9$

d)

$\frac{64xy}{3}$

7.

Simplify by first converting to rational exponents. Assume that all variables represent positive real numbers.

$11\sqrt[3]{2}-4\sqrt[3]{128}$

a)

$-5\sqrt[3]{2}$

b)

$5\sqrt[3]{2}$

c)

$\sqrt[3]{5}$

d)

$11\sqrt[3]{2}-4\sqrt[3]{128}$

8.

Write with radicals. Assume that all variables represent positive real numbers.

$\left(8x\right)^{\frac{4}{7}}+\left(3y\right)^{\frac{4}{5}}$

a)

$\left(\sqrt[7]{8}\right)^4+\left(\sqrt[5]{3}\right)^4$

b)

$\left(\sqrt[7]{8}\right)^4x+\left(\sqrt[5]{3}\right)^4$

c)

$\left(\sqrt[4]{8x}\right)^7+\left(\sqrt[4]{3y}\right)^5$

d)

$\left(\sqrt[7]{8x}\right)^4+\left(\sqrt[5]{3y}\right)^4$

9.

Simplify by first converting to rational exponents. Assume that all variables represent positive real numbers.

$\frac{\sqrt[5]{w^6}}{\sqrt[7]{w^8}}$

a)

$\sqrt[12]{w^8}$

b)

$\sqrt[35]{w^2}$

c)

$\sqrt[2]{w^8}$

d)

$\sqrt[35]{2w^8}$

10.

Use the rules of exponents to simplify the expression. Write the answer with positive exponents. Assume that all variables represent positive real numbers

$\left(\frac{x^2}{y^{-4}}\right)^{\frac{1}{2}}$

a)

$xy$

b)

$\frac{x}{y^2}$

c)

$xy^{\frac{1}{2}}$

d)

$xy^2$

11.

Use the rules of exponents to simplify the expression. Write the answer with positive exponents. Assume that all variables represent positive real numbers

$\left(256\frac{a^3b^{-5}}{a^{-1}b^3}\right)^{\frac{1}{4}}$

a)

$\frac{16b^2}{a}$

b)

$\frac{16a}{b^2}$

c)

$\frac{4a}{b^2}$

d)

$4ab^2$

12.

Write with rational exponents and then apply the properties of exponents. Assume that all radicands represent positive real numbers. Give answers in exponential form.

$\frac{\sqrt[]{x^5}}{\sqrt[]{x^{10}}}$

a)

$\frac{1}{x^{\frac{15}{2}}}$

b)

$x^{\frac{5}{2}}$

c)

$\frac{1}{x^{\frac{5}{2}}}$

d)

$\frac{1}{x^{10}}$

13.

Write with rational exponents and then apply the properties of exponents. Assume that all radicands represent positive real numbers. Give answers in exponential form.

$\sqrt[5]{\sqrt[4]{w}}$

a)

$\frac{1}{w^{20}}$

b)

$20w$

c)

$w^{20}$

d)

$w^{\frac{1}{20}}$

14.

Use the rules of exponents to simplify the expression. Write the answer with positive exponents. Assume that all variables represent positive real numbers

$\left(256\frac{a^3b^{-5}}{a^{-1}b^3}\right)^{\frac{1}{4}}$

a)

$\frac{16b^2}{a}$

b)

$\frac{16a}{b^2}$

c)

$\frac{4a}{b^2}$

d)

$4ab^2$

15.

Use the rules of exponents to simplify the expression. Write the answer with positive exponents. Assume that all variables represent positive real numbers

$\frac{x^{\frac{1}{2}}x^{\frac{3}{10}}x^{\frac{2}{5}}}{\left(x^2\right)^{-\frac{1}{2}}}$

a)

$\frac{1}{x^{\frac{11}{5}}}$

b)

$x^{\frac{22}{7}}$

c)

$\frac{1}{x^{\frac{22}{7}}}$

d)

$x^{\frac{11}{5}}$

16.

Rationalize the denominator. Assume that all variables represent positive real numbers and that the denominator is not zero.

$\frac{5}{7-\sqrt[]{2}}$

a)

$\frac{5}{7}-\frac{5}{\sqrt[]{2}}$

b)

$\frac{35-5\sqrt[]{2}}{47}$

c)

$\frac{35+5\sqrt[]{2}}{5}$

d)

$\frac{35+5\sqrt[]{2}}{47}$

17.

Rationalize the denominator. Assume that all variables represent positive real numbers and that the denominator is not zero.

$\frac{5-\sqrt[]{2}}{5+\sqrt[]{2}}$

a)

$-1$

b)

$\frac{27-10\sqrt[]{2}}{23}$

c)

$\frac{27+10\sqrt[]{2}}{23}$

d)

$\frac{23-10\sqrt[]{2}}{27}$

18.

Perform the indicated operation. Assume all variables are positive.

$\sqrt[4]{16w^{10}}\ +\ 2w\sqrt[4]{w^6}$

a)

$4w^2\left(\sqrt[4]{w^2}\right)$

b)

$2w^2\left(\sqrt[4]{w^2}\right)$

c)

$4w^{ }\left(\sqrt[4]{w^2}\right)$

d)

$4w^2$

19.

Find simplified expressions for the area of the figure.

a)

$6x^{\frac{2}{3}}$

b)

$6x^{\frac{1}{3}}$

c)

$12x^{\frac{2}{3}}$

d)

$12x^{\frac{1}{3}}$

20.

Use the properties of radicals to simplify the expression.

$\frac{\left(\sqrt[3]{6}\right)\left(\sqrt[3]{72}\right)}{\sqrt[3]{2}}$

a)

$6$

b)

$216$

c)

$72$

d)

$144$

21.

Simplify the expression

$\sqrt[5]{224}+3\sqrt[5]{7}$

a)

$5\sqrt[]{7}$

b)

$5\sqrt[5]{7}$

c)

$5\sqrt[5]{14}$

d)

$\sqrt[5]{7}$

22.

Explain how to evaluate $\left(-64\right)^{\frac{5}{3}}$

4 lines

23.

What is wrong with this?

$\left(4x^3\right)^{-1}=4x^{-3}$

4 lines

24.

How many real square roots does any positive number have?

4 lines

25.

Explain why $\sqrt[3]{-5}$ and $-\sqrt[3]{5}$ represent the same number.

4 lines

Rational Exponents and Radicals

Total questions: 25

Worksheet time: 1hrs 13mins

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