QuestionSimplify $\sqrt[4]{16 x^{12}}$ for $x>0$ .

Studdy Solution

STEP 1

Assumptions1. $x$ is a positive real number. $\sqrt[4]{16 x^{12}}$ is a fourth root expression that needs to be simplified

STEP 2

We can simplify the expression by taking the fourth root of each factor separately.
$\sqrt[4]{16 x^{12}} = \sqrt[4]{16} \cdot \sqrt[4]{x^{12}}$

STEP 3

Calculate the fourth root of16.
$\sqrt[]{16} =2$

STEP 4

For the second part of the expression, we can rewrite $x^{12}$ as $(x^3)^4$ .
$\sqrt[4]{x^{12}} = \sqrt[4]{(x^3)^4}$

STEP 5

The fourth root of $(x^3)^4$ is $x^3$ because the fourth root and the fourth power cancel each other out.
$\sqrt[4]{(x^3)^4} = x^3$

STEP 6

Now, multiply the results from steps3 and5 to get the final simplified expression.
$\sqrt[4]{16 x^{12}} =2 \cdot x^3$

STEP 7

implify the expression.
$\sqrt[4]{16 x^{12}} =2x^3$ So, $\sqrt[4]{16 x^{12}}$ simplifies to $2x^3$ .

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