Question

which of the following are equivalent to the expression below?
choose two correct answers.
$\boldsymbol{ -\frac{4}{\sqrt3{64}}}$
\\(\square\\) a. \\(-\dfrac{1}{16}\\)
\\(\square\\) b. \\(-16\\)
\\(\square\\) c. \\(-1\\)
\\(\square\\) d. \\(-4(64)^{\frac{1}{3}}\\)
\\(\square\\) e. \\(-4(64)^{-\frac{1}{3}}\\)

Explanation:

Step1: Simplify the cube root of 64

We know that \( \sqrt[3]{64} = 4 \) because \( 4\times4\times4 = 64 \). So the original expression \( -\frac{4}{\sqrt[3]{64}} \) becomes \( -\frac{4}{4} \).

Step2: Simplify the fraction

Simplifying \( -\frac{4}{4} \) gives \( - 1 \), so the original expression is equal to \( - 1 \), which corresponds to option C.

Step3: Recall the negative exponent rule

The negative exponent rule states that \( a^{-n}=\frac{1}{a^{n}} \), so \( \frac{1}{\sqrt[3]{64}}=\frac{1}{64^{\frac{1}{3}}} = 64^{-\frac{1}{3}} \). Then the original expression \( -\frac{4}{\sqrt[3]{64}} \) can be rewritten as \( - 4\times64^{-\frac{1}{3}} \), which corresponds to option E.

Step4: Check other options

• Option A: \( -\frac{1}{16}

eq - 1 \), so A is incorrect.

• Option B: \( - 16

eq - 1 \), so B is incorrect.

• Option D: \( - 4(64)^{\frac{1}{3}}=-4\times4 = - 16

eq - 1 \), so D is incorrect.

Answer:

C. \(-1\)
E. \(-4(64)^{-1/3}\)