Algebra Examples

$\frac{{(6 x^{4})}^{- 2}}{{(\frac{x^{- 2} \sqrt{2}}{x^{3} \sqrt{16}})}^{2}}$

Step 1

Reduce the expression by cancelling the common factors.

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Step 1.1

Move ${(6 x^{4})}^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{{(\frac{x^{- 2} \sqrt{2}}{x^{3} \sqrt{16}})}^{2} {(6 x^{4})}^{2}}$

Step 1.2

Cancel the common factor of $x^{- 2}$ and $x^{3}$ .

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Step 1.2.1

Factor $x^{- 2}$ out of $x^{- 2} \sqrt{2}$ .

$\frac{1}{{(\frac{x^{- 2} (\sqrt{2})}{x^{3} \sqrt{16}})}^{2} {(6 x^{4})}^{2}}$

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Factor $x^{- 2}$ out of $x^{3} \sqrt{16}$ .

$\frac{1}{{(\frac{x^{- 2} (\sqrt{2})}{x^{- 2} (x^{5} \sqrt{16})})}^{2} {(6 x^{4})}^{2}}$

Step 1.2.2.2

Cancel the common factor.

$\frac{1}{{(\frac{x^{- 2} \sqrt{2}}{x^{- 2} (x^{5} \sqrt{16})})}^{2} {(6 x^{4})}^{2}}$

Step 1.2.2.3

Rewrite the expression.

$\frac{1}{{(\frac{\sqrt{2}}{x^{5} \sqrt{16}})}^{2} {(6 x^{4})}^{2}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $16$ as $4^{2}$ .

$\frac{1}{{(\frac{\sqrt{2}}{x^{5} \sqrt{4^{2}}})}^{2} {(6 x^{4})}^{2}}$

Step 2.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{1}{{(\frac{\sqrt{2}}{x^{5} \cdot 4})}^{2} {(6 x^{4})}^{2}}$

Step 3

Simplify the denominator.

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Step 3.1

Apply the product rule to $\frac{\sqrt{2}}{x^{5} \cdot 4}$ .

$\frac{1}{\frac{{\sqrt{2}}^{2}}{{(x^{5} \cdot 4)}^{2}} {(6 x^{4})}^{2}}$

Step 3.2

Apply the product rule to $6 x^{4}$ .

$\frac{1}{\frac{{\sqrt{2}}^{2}}{{(x^{5} \cdot 4)}^{2}} \cdot 6^{2} {(x^{4})}^{2}}$

Step 3.3

Combine exponents.

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Step 3.3.1

Combine $\frac{{\sqrt{2}}^{2}}{{(x^{5} \cdot 4)}^{2}}$ and $6^{2}$ .

$\frac{1}{\frac{{\sqrt{2}}^{2} \cdot 6^{2}}{{(x^{5} \cdot 4)}^{2}} {(x^{4})}^{2}}$

Step 3.3.2

Combine $\frac{{\sqrt{2}}^{2} \cdot 6^{2}}{{(x^{5} \cdot 4)}^{2}}$ and ${(x^{4})}^{2}$ .

$\frac{1}{\frac{{\sqrt{2}}^{2} \cdot 6^{2} {(x^{4})}^{2}}{{(x^{5} \cdot 4)}^{2}}}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 3.4.1.1

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

$\frac{1}{\frac{{(2^{\frac{1}{2}})}^{2} \cdot 6^{2} {(x^{4})}^{2}}{{(x^{5} \cdot 4)}^{2}}}$

Step 3.4.1.2

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

$\frac{1}{\frac{2^{\frac{1}{2} \cdot 2} \cdot 6^{2} {(x^{4})}^{2}}{{(x^{5} \cdot 4)}^{2}}}$

Step 3.4.1.3

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

$\frac{1}{\frac{2^{\frac{2}{2}} \cdot 6^{2} {(x^{4})}^{2}}{{(x^{5} \cdot 4)}^{2}}}$

Step 3.4.1.4

Cancel the common factor of $2$ .

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Step 3.4.1.4.1

Cancel the common factor.

$\frac{1}{\frac{2^{\frac{2}{2}} \cdot 6^{2} {(x^{4})}^{2}}{{(x^{5} \cdot 4)}^{2}}}$

Step 3.4.1.4.2

Rewrite the expression.

$\frac{1}{\frac{2^{1} \cdot 6^{2} {(x^{4})}^{2}}{{(x^{5} \cdot 4)}^{2}}}$

Step 3.4.1.5

Evaluate the exponent.

$\frac{1}{\frac{2 \cdot 6^{2} {(x^{4})}^{2}}{{(x^{5} \cdot 4)}^{2}}}$

Step 3.4.2

Raise $6$ to the power of $2$ .

$\frac{1}{\frac{2 \cdot 36 {(x^{4})}^{2}}{{(x^{5} \cdot 4)}^{2}}}$

Step 3.4.3

Multiply the exponents in ${(x^{4})}^{2}$ .

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Step 3.4.3.1

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

$\frac{1}{\frac{2 \cdot 36 x^{4 \cdot 2}}{{(x^{5} \cdot 4)}^{2}}}$

Step 3.4.3.2

Multiply $4$ by $2$ .

$\frac{1}{\frac{2 \cdot 36 x^{8}}{{(x^{5} \cdot 4)}^{2}}}$

Step 3.4.4

Multiply $2$ by $36$ .

$\frac{1}{\frac{72 x^{8}}{{(x^{5} \cdot 4)}^{2}}}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Apply the product rule to $x^{5} \cdot 4$ .

$\frac{1}{\frac{72 x^{8}}{{(x^{5})}^{2} \cdot 4^{2}}}$

Step 3.5.2

Multiply the exponents in ${(x^{5})}^{2}$ .

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Step 3.5.2.1

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

$\frac{1}{\frac{72 x^{8}}{x^{5 \cdot 2} \cdot 4^{2}}}$

Step 3.5.2.2

Multiply $5$ by $2$ .

$\frac{1}{\frac{72 x^{8}}{x^{10} \cdot 4^{2}}}$

Step 3.5.3

Raise $4$ to the power of $2$ .

$\frac{1}{\frac{72 x^{8}}{x^{10} \cdot 16}}$

Step 3.6

Cancel the common factor of $72$ and $16$ .

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Step 3.6.1

Factor $8$ out of $72 x^{8}$ .

$\frac{1}{\frac{8 (9 x^{8})}{x^{10} \cdot 16}}$

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

Factor $8$ out of $x^{10} \cdot 16$ .

$\frac{1}{\frac{8 (9 x^{8})}{8 (x^{10} \cdot 2)}}$

Step 3.6.2.2

Cancel the common factor.

$\frac{1}{\frac{8 (9 x^{8})}{8 (x^{10} \cdot 2)}}$

Step 3.6.2.3

Rewrite the expression.

$\frac{1}{\frac{9 x^{8}}{x^{10} \cdot 2}}$

Step 3.7

Cancel the common factor of $x^{8}$ and $x^{10}$ .

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Step 3.7.1

Factor $x^{8}$ out of $9 x^{8}$ .

$\frac{1}{\frac{x^{8} \cdot 9}{x^{10} \cdot 2}}$

Step 3.7.2

Cancel the common factors.

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Step 3.7.2.1

Factor $x^{8}$ out of $x^{10} \cdot 2$ .

$\frac{1}{\frac{x^{8} \cdot 9}{x^{8} (x^{2} \cdot 2)}}$

Step 3.7.2.2

Cancel the common factor.

$\frac{1}{\frac{x^{8} \cdot 9}{x^{8} (x^{2} \cdot 2)}}$

Step 3.7.2.3

Rewrite the expression.

$\frac{1}{\frac{9}{x^{2} \cdot 2}}$

Step 3.8

Move $2$ to the left of $x^{2}$ .

$\frac{1}{\frac{9}{2 x^{2}}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{2 x^{2}}{9}$

Step 5

Multiply $\frac{2 x^{2}}{9}$ by $1$ .

$\frac{2 x^{2}}{9}$