QuestionSimplify $\sqrt[3]{27 x^{21}}$ for $x>0$ .

Studdy Solution

STEP 1

Assumptions1. The given expression is $\sqrt[3]{27 x^{21}}$ . . The variable $x$ is greater than zero.

STEP 2

We need to simplify the given expression. The cube root of a number $a$ is a value that, when cubed, gives the number $a$ . We can write the cube root of a number as the number raised to the power of $\frac{1}{}$ .
$\sqrt[]{27 x^{21}} = (27 x^{21})^{\frac{1}{}}$

STEP 3

We can simplify this further by applying the rule of exponents which states that $(a^m)^n = a^{mn}$ .
$(27 x^{21})^{\frac{1}{3}} =27^{\frac{1}{3}} \times (x^{21})^{\frac{1}{3}}$

STEP 4

Now, simplify the expression $27^{\frac{1}{3}}$ and $(x^{21})^{\frac{1}{3}}$ separately.
$27^{\frac{1}{3}} =3$ $(x^{21})^{\frac{1}{3}} = x^{21 \times \frac{1}{3}} = x^7$

STEP 5

Substitute these values back into the equation.
$27^{\frac{1}{3}} \times (x^{21})^{\frac{1}{3}} =3 \times x^7$ So, the simplified form of $\sqrt[3]{27 x^{21}}$ is $3x^7$ .

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