Exercises: Explain How the Definition of Rational Exponents Follows from Integer Exponent Properties

Work through each section in order. Show your reasoning where indicated. For explanation problems, write in complete sentences.

Grade 9·20 problems·~30 min·Common Core Math - HS Number and Quantity·standard·hsn-rn-a-1

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Warm-Up: Integer Exponent Rules

These problems review skills you already know.

The power rule states that $(a^m)^n = a^{mn}$ . Applying this rule to $(5^{1/3})^3$ gives $5^{(1/3) \cdot 3} = 5^1 = 5$ . Which conclusion follows?

$5^{1/3}$ must be the number whose cube is 5, so $5^{1/3} = \sqrt[3]{5}$

$5^{1/3}$ must equal $\frac{5}{3}$ , since the exponent denominator is 3

$5^{1/3}$ must equal $\frac{1}{5^3}$ , since fractional exponents produce reciprocals

$5^{1/3}$ must equal $5^3$ , since roots and powers are inverse operations

Which statement correctly converts between radical and exponential notation?

$\sqrt[4]{x} = x^4$

$\sqrt[4]{x} = x^{1/4}$

$\sqrt[4]{x} = 4x$

$\sqrt[4]{x} = x^{-4}$

The product rule states $a^m \cdot a^n = a^{m+n}$ . Simplify $4^{1/2} \cdot 4^{3/2}$ using this rule. Show each step and state which rule you used.

Evaluate each expression.

What is the value of $8^{1/3}$ ?

$\frac{8}{3}$

$2$

$24$

$\sqrt{8}$

Evaluate $16^{3/4}$ .

Evaluate $27^{2/3}$ .

Simplify $27^{2/3} \div 27^{1/3}$ using the quotient rule $a^r \div a^s = a^{r-s}$ . Give a whole number answer.

Evaluate $8^{-2/3}$ . Express your answer as a fraction.

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