Question

5 multiple answer 1 point choose all the expressions that are equivalent to: $27^{\frac{4}{3}}$ 18 $\sqrt4{27^{3}}$ $9\sqrt3{27}$ $\sqrt3{27^{4}}$ 81

Explanation:

Step1: Simplify the original expression

First, rewrite 27 as $3^3$:
$$27^{\frac{4}{3}} = (3^3)^{\frac{4}{3}}$$
Use exponent rule $(a^m)^n = a^{m \cdot n}$:
$$(3^3)^{\frac{4}{3}} = 3^{3 \times \frac{4}{3}} = 3^4 = 81$$

Step2: Analyze each option

Option 1: 18

18 ≠ 81, so not equivalent.

Option 2: $\sqrt[4]{27^3}$

Rewrite as exponent form: $27^{\frac{3}{4}}$. This is not equal to $27^{\frac{4}{3}}$, so not equivalent.

Option 3: $9\sqrt[3]{27}$

Simplify $\sqrt[3]{27}=3$, so $9 \times 3 = 27$. 27 ≠ 81, so not equivalent.

Option 4: $\sqrt[3]{27^4}$

Rewrite as exponent form: $27^{\frac{4}{3}}$, which matches the original expression. Simplify:
$$\sqrt[3]{27^4} = (3^3)^{\frac{4}{3}} = 3^4 = 81$$
Equivalent to original.

Option 5: 81

Matches the simplified value of the original expression, so equivalent.

Answer:

$\sqrt[3]{27^4}$, 81