Algebra Examples

${(\sqrt{27 x^{9}})}^{4}$

Step 1

Rewrite $27 x^{9}$ as ${(3 x^{4})}^{2} \cdot (3 x)$ .

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Step 1.1

Factor $9$ out of $27$ .

${\sqrt{9 (3) x^{9}}}^{4}$

Step 1.2

Rewrite $9$ as $3^{2}$ .

${\sqrt{3^{2} \cdot 3 x^{9}}}^{4}$

Step 1.3

Factor out $x^{8}$ .

${\sqrt{3^{2} \cdot 3 (x^{8} x)}}^{4}$

Step 1.4

Rewrite $x^{8}$ as ${(x^{4})}^{2}$ .

${\sqrt{3^{2} \cdot 3 ({(x^{4})}^{2} x)}}^{4}$

Step 1.5

Move $3$ .

${\sqrt{3^{2} {(x^{4})}^{2} \cdot 3 x}}^{4}$

Step 1.6

Rewrite $3^{2} {(x^{4})}^{2}$ as ${(3 x^{4})}^{2}$ .

${\sqrt{{(3 x^{4})}^{2} \cdot 3 x}}^{4}$

Step 1.7

Add parentheses.

${\sqrt{{(3 x^{4})}^{2} \cdot (3 x)}}^{4}$

Step 2

Pull terms out from under the radical.

${(3 x^{4} \sqrt{3 x})}^{4}$

Step 3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1

Apply the product rule to $3 x^{4} \sqrt{3 x}$ .

${(3 x^{4})}^{4} {\sqrt{3 x}}^{4}$

Step 3.2

Apply the product rule to $3 x^{4}$ .

$3^{4} {(x^{4})}^{4} {\sqrt{3 x}}^{4}$

Step 4

Simplify the expression.

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Step 4.1

Raise $3$ to the power of $4$ .

$81 {(x^{4})}^{4} {\sqrt{3 x}}^{4}$

Step 4.2

Multiply the exponents in ${(x^{4})}^{4}$ .

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Step 4.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$81 x^{4 \cdot 4} {\sqrt{3 x}}^{4}$

Step 4.2.2

Multiply $4$ by $4$ .

$81 x^{16} {\sqrt{3 x}}^{4}$

Step 5

Rewrite ${\sqrt{3 x}}^{4}$ as ${(3 x)}^{2}$ .

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Step 5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x}$ as ${(3 x)}^{\frac{1}{2}}$ .

$81 x^{16} {({(3 x)}^{\frac{1}{2}})}^{4}$

Step 5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$81 x^{16} {(3 x)}^{\frac{1}{2} \cdot 4}$

Step 5.3

Combine $\frac{1}{2}$ and $4$ .

$81 x^{16} {(3 x)}^{\frac{4}{2}}$

Step 5.4

Cancel the common factor of $4$ and $2$ .

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Step 5.4.1

Factor $2$ out of $4$ .

$81 x^{16} {(3 x)}^{\frac{2 \cdot 2}{2}}$

Step 5.4.2

Cancel the common factors.

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Step 5.4.2.1

Factor $2$ out of $2$ .

$81 x^{16} {(3 x)}^{\frac{2 \cdot 2}{2 (1)}}$

Step 5.4.2.2

Cancel the common factor.

$81 x^{16} {(3 x)}^{\frac{2 \cdot 2}{2 \cdot 1}}$

Step 5.4.2.3

Rewrite the expression.

$81 x^{16} {(3 x)}^{\frac{2}{1}}$

Step 5.4.2.4

Divide $2$ by $1$ .

$81 x^{16} {(3 x)}^{2}$

Step 6

Simplify the expression.

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Step 6.1

Apply the product rule to $3 x$ .

$81 x^{16} (3^{2} x^{2})$

Step 6.2

Rewrite using the commutative property of multiplication.

$81 \cdot 3^{2} x^{16} x^{2}$

Step 7

Multiply $x^{16}$ by $x^{2}$ by adding the exponents.

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Step 7.1

Move $x^{2}$ .

$81 \cdot 3^{2} (x^{2} x^{16})$

Step 7.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$81 \cdot 3^{2} x^{2 + 16}$

Step 7.3

Add $2$ and $16$ .

$81 \cdot 3^{2} x^{18}$

Step 8

Raise $3$ to the power of $2$ .

$81 \cdot 9 x^{18}$

Step 9

Multiply $81$ by $9$ .

$729 x^{18}$