Question

\\\frac{1}{(\sqrt{3k})^5}\\

Explanation:

⚡ Using what you learned: Rationalizing Denominators · Expressions with Negative Exponents

Step 1: Rewrite using rational exponents

$$ \frac{1}{(\sqrt{3k})^5} = \frac{1}{(3k)^{5/2}} = (3k)^{-5/2} $$

Step 2: Simplify the radical in the denominator

$$ (\sqrt{3k})^5 = (\sqrt{3k})^4 \cdot \sqrt{3k} = (3k)^2 \sqrt{3k} = 9k^2 \sqrt{3k} $$

$$ \frac{1}{(\sqrt{3k})^5} = \frac{1}{9k^2 \sqrt{3k}} $$

Step 3: Rationalize the denominator

Multiply the numerator and denominator by \(\sqrt{3k}\):

$$ \frac{1 \cdot \sqrt{3k}}{9k^2 \sqrt{3k} \cdot \sqrt{3k}} = \frac{\sqrt{3k}}{9k^2 (3k)} = \frac{\sqrt{3k}}{27k^3} $$

Answer:

The expression can be written in the following equivalent forms:

In exponential form:

$$ (3k)^{-5/2} $$

With a rationalized denominator:

$$ \frac{\sqrt{3k}}{27k^3} $$