Questions: Simplify -7 ∛(625 x^8) completely. Write answer in radical form.

Transcript text: Simplify $-7 \sqrt[3]{625 x^{8}}$ completely. Write answer in radical form.

Solution

Solution Steps

Step 1: Factor the radicand

Factor $ 625x^8 $ into its prime factors: \[ 625 = 5^4, \quad x^8 = x^{6} \cdot x^{2}. \] Thus, the expression becomes: \[ -7 \sqrt[3]{5^4 x^6 x^2}. \]

Step 2: Simplify the cube root

Extract the cube roots of the perfect cubes: \[ \sqrt[3]{5^4 x^6 x^2} = \sqrt[3]{5^3 \cdot 5 \cdot x^6 \cdot x^2} = 5x^2 \sqrt[3]{5x^2}. \]

Step 3: Combine the terms

Multiply the extracted terms with the coefficient $-7$: \[ -7 \cdot 5x^2 \sqrt[3]{5x^2} = -35x^2 \sqrt[3]{5x^2}. \]

Final Answer

$\boxed{-35x^2 \sqrt[3]{5x^2}}$

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