Question

drag the radical expressions given above to the appropriate box so they match the expression with rational exponents to which they are equivalent. choices may be used once, more than once, or not at all.
(a) (-6x^{\frac{1}{3}})
(b) (-6x^{-\frac{1}{3}})
(c) (6x^{-\frac{1}{3}})
(d) (6x^{\frac{1}{3}})

Explanation:

Step1: Recall radical - exponent rule

The rule is $a^{\frac{m}{n}}=\sqrt[n]{a^{m}}$ and $a^{-\frac{m}{n}}=\frac{1}{\sqrt[n]{a^{m}}}$.

Step2: Analyze (a) $- 6x^{\frac{1}{3}}$

Using the rule $a^{\frac{m}{n}}=\sqrt[n]{a^{m}}$, we have $x^{\frac{1}{3}}=\sqrt[3]{x}$, so $-6x^{\frac{1}{3}}=-6\sqrt[3]{x}$.

Step3: Analyze (b) $-6x^{-\frac{1}{3}}$

Using the rule $a^{-\frac{m}{n}}=\frac{1}{\sqrt[n]{a^{m}}}$, we get $x^{-\frac{1}{3}}=\frac{1}{\sqrt[3]{x}}$, so $-6x^{-\frac{1}{3}}=-\frac{6}{\sqrt[3]{x}}$.

Step4: Analyze (c) $6x^{-\frac{1}{3}}$

Since $x^{-\frac{1}{3}}=\frac{1}{\sqrt[3]{x}}$, then $6x^{-\frac{1}{3}}=\frac{6}{\sqrt[3]{x}}$.

Step5: Analyze (d) $6x^{\frac{1}{3}}$

As $x^{\frac{1}{3}}=\sqrt[3]{x}$, so $6x^{\frac{1}{3}} = 6\sqrt[3]{x}$ (not given in the options).

Answer:

(a) matches $-6\sqrt[3]{x}$
(b) matches $-\frac{6}{\sqrt[3]{x}}$
(c) matches $\frac{6}{\sqrt[3]{x}}$
(d) has no match among the given options.