Questions: Simplify each radical expression. (a) ∛(16 x^5) (b) √(45 x^2 y^(-4))

Simplify each radical expression. (a) ∛(16 x^5) (b) √(45 x^2 y^(-4))
Transcript text: Simplify each radical expression. (a) $\sqrt[3]{16 x^{5}}$ (b) $\sqrt{45 x^{2} y^{-4}}$
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Solution

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Solution Steps

Step 1: Simplifying \( \sqrt[3]{16 x^{5}} \)

We start with the expression \( \sqrt[3]{16 x^{5}} \). We can express 16 as \( 2^4 \), so we rewrite the expression as: \[ \sqrt[3]{2^4 \cdot x^5} = \sqrt[3]{2^4} \cdot \sqrt[3]{x^5} \] This simplifies to: \[ 2 \cdot \sqrt[3]{2} \cdot x^{5/3} = 2x \cdot \sqrt[3]{2} \cdot x^{2/3} = 2x \sqrt[3]{2 x^{2}} \]

Step 2: Simplifying \( \sqrt{45 x^{2} y^{-4}} \)

Next, we simplify the expression \( \sqrt{45 x^{2} y^{-4}} \). We can express 45 as \( 3^2 \cdot 5 \), so we rewrite the expression as: \[ \sqrt{3^2 \cdot 5 \cdot x^2 \cdot y^{-4}} = \sqrt{3^2} \cdot \sqrt{5} \cdot \sqrt{x^2} \cdot \sqrt{y^{-4}} \] This simplifies to: \[ 3 \cdot \sqrt{5} \cdot x \cdot \frac{1}{y^2} = \frac{3x \sqrt{5}}{y^2} \]

Final Answer

The simplified forms of the radical expressions are:

  1. For \( \sqrt[3]{16 x^{5}} \): \( 2x \sqrt[3]{2 x^{2}} \)
  2. For \( \sqrt{45 x^{2} y^{-4}} \): \( \frac{3x \sqrt{5}}{y^2} \)

Thus, the final answers are: \[ \boxed{2x \sqrt[3]{2 x^{2}}} \] \[ \boxed{\frac{3x \sqrt{5}}{y^2}} \]

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