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Radicals and Exponents Practice

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Radicals and Exponents Practice

Rewrite radicals using rational exponents, apply exponent rules, and simplify. Assume all variables are positive. Show your work where space is provided.

Rewrite Radicals and Rational Exponents

Convert between radical form and rational exponent form.

1.Write ∛(x⁵) using a rational exponent.

2.Rewrite p^(7/3) in radical form.

3.Which rational exponent is equivalent to ⁵√(q³)?

q^(3/5)
q^(5/3)
q^(1/15)
q^15

4.Simplify √(49r⁶) by writing the radical as a rational exponent first.

5.Match each radical expression to its equivalent rational exponent form.

⁴√(a³)
∛(b⁵)
⁶√(c⁴)
√(d⁷)

a^(3/4)
b^(5/3)
d^(7/2)
c^(2/3)

Use the Exponent Rules

Simplify each expression. Combine exponents carefully.

6.Simplify u^(2/7) × u^(5/7).

7.Simplify k^(11/6) ÷ k^(5/6).

8.Simplify (t^(3/8))⁴.

9.Simplify (v^(1/3) × v^(1/6))².

10.Simplify (x^(5/4) × x^(3/4)) ÷ x^(1/2).

Mixed Practice

Use the forms and rules you have practiced.

11.Which expression is not equivalent to the other three?

√n
n^(1/2)
n^(2/4)
n²

12.A fire station stores water in a spherical tank with volume 36π cubic feet. Use r = (3V/(4π))^(1/3) to write the tank's radius in radical form and find its value.

13.Order the exponents 1/4, 3/4, 1/2, and 1 from least to greatest. Use the number line to help.