Exercises: Rewrite Expressions Involving Radicals and Rational Exponents

Work through each section in order. Show your work for simplification problems. Express all fractional exponents in lowest terms.

Grade 9·20 problems·~28 min·Common Core Math - HS Number and Quantity·standard·hsn-rn-a-2

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Warm-Up: The Notation Bridge

These problems review the basic conversion between radical and exponential notation.

Which expression is equivalent to $\sqrt[5]{x^3}$ ?

$x^{5/3}$

$x^{3/5}$

$5x^3$

$x^{15}$

Which radical expression is equivalent to $y^{4/7}$ ?

$\sqrt[4]{y^7}$

$\sqrt[7]{y^4}$

$7y^4$

$(\sqrt[4]{y})^7$

Evaluate $\sqrt[3]{8x^3}$ by first writing it in exponential form as $(8x^3)^{1/3}$ , then simplifying. Give your answer as a simplified expression.

Rewrite each expression as indicated. Simplify completely.

Rewrite each expression in exponential form (as $a^{m/n}$ ):

(a) $\sqrt[4]{x^3}$

(b) $\sqrt{x^7}$

Which expression correctly converts $z^{5/4}$ to radical notation?

$\sqrt[5]{z^4}$

$\sqrt[4]{z^5}$

$(\sqrt[5]{z})^4$

$\sqrt{z^{5/4}}$

Rewrite each expression in radical notation:

(a) $a^{3/5}$

(b) $b^{-1/2}$

Simplify using the product rule $a^r \cdot a^s = a^{r+s}$ :

(a) $x^{1/2} \cdot x^{1/3}$

(b) $x^{1/4} \cdot x^{3/4}$

Simplify using the power rule $(a^r)^s = a^{rs}$ or the quotient rule $a^r / a^s = a^{r-s}$ :

(a) $(x^{2/3})^{3/4}$

(b) $x^{3/4} \div x^{1/4}$

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