Algebra Examples

$\frac{6 \sqrt{x^{2}} \sqrt{x^{2}}}{81 \sqrt{x^{16}}}$

Step 1

Cancel the common factor of $6$ and $81$ .

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Step 1.1

Factor $3$ out of $6 \sqrt{x^{2}} \sqrt{x^{2}}$ .

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

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $3$ out of $81 \sqrt{x^{16}}$ .

$\frac{3 (2 \sqrt{x^{2}} \sqrt{x^{2}})}{3 (27 \sqrt{x^{16}})}$

Step 1.2.2

Cancel the common factor.

$\frac{3 (2 \sqrt{x^{2}} \sqrt{x^{2}})}{3 (27 \sqrt{x^{16}})}$

Step 1.2.3

Rewrite the expression.

$\frac{2 \sqrt{x^{2}} \sqrt{x^{2}}}{27 \sqrt{x^{16}}}$

Step 2

Combine $\sqrt{x^{2}}$ and $\sqrt{x^{16}}$ into a single radical.

$\frac{2 \sqrt{\frac{x^{2}}{x^{16}}} \sqrt{x^{2}}}{27}$

Step 3

Cancel the common factor of $x^{2}$ and $x^{16}$ .

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Step 3.1

Multiply by $1$ .

$\frac{2 \sqrt{\frac{x^{2} \cdot 1}{x^{16}}} \sqrt{x^{2}}}{27}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $x^{2}$ out of $x^{16}$ .

$\frac{2 \sqrt{\frac{x^{2} \cdot 1}{x^{2} x^{14}}} \sqrt{x^{2}}}{27}$

Step 3.2.2

Cancel the common factor.

$\frac{2 \sqrt{\frac{x^{2} \cdot 1}{x^{2} x^{14}}} \sqrt{x^{2}}}{27}$

Step 3.2.3

Rewrite the expression.

$\frac{2 \sqrt{\frac{1}{x^{14}}} \sqrt{x^{2}}}{27}$

Step 4

Simplify the numerator.

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Step 4.1

Combine using the product rule for radicals.

$\frac{2 \sqrt{x^{2} \frac{1}{x^{14}}}}{27}$

Step 4.2

Combine $x^{2}$ and $\frac{1}{x^{14}}$ .

$\frac{2 \sqrt{\frac{x^{2}}{x^{14}}}}{27}$

Step 5

Simplify the numerator.

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Step 5.1

Reduce the expression $\frac{x^{2}}{x^{14}}$ by cancelling the common factors.

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Step 5.1.1

Multiply by $1$ .

$\frac{2 \sqrt{\frac{x^{2} \cdot 1}{x^{14}}}}{27}$

Step 5.1.2

Factor $x^{2}$ out of $x^{14}$ .

$\frac{2 \sqrt{\frac{x^{2} \cdot 1}{x^{2} x^{12}}}}{27}$

Step 5.1.3

Cancel the common factor.

$\frac{2 \sqrt{\frac{x^{2} \cdot 1}{x^{2} x^{12}}}}{27}$

Step 5.1.4

Rewrite the expression.

$\frac{2 \sqrt{\frac{1}{x^{12}}}}{27}$

Step 5.2

Rewrite $1$ as $1^{2}$ .

$\frac{2 \sqrt{\frac{1^{2}}{x^{12}}}}{27}$

Step 5.3

Rewrite $x^{12}$ as ${(x^{6})}^{2}$ .

$\frac{2 \sqrt{\frac{1^{2}}{{(x^{6})}^{2}}}}{27}$

Step 5.4

Rewrite $\frac{1^{2}}{{(x^{6})}^{2}}$ as ${(\frac{1}{x^{6}})}^{2}$ .

$\frac{2 \sqrt{{(\frac{1}{x^{6}})}^{2}}}{27}$

Step 5.5

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

$\frac{2 \frac{1}{x^{6}}}{27}$

Step 6

Combine $2$ and $\frac{1}{x^{6}}$ .

$\frac{\frac{2}{x^{6}}}{27}$

Step 7

Multiply the numerator by the reciprocal of the denominator.

$\frac{2}{x^{6}} \cdot \frac{1}{27}$

Step 8

Combine.

$\frac{2 \cdot 1}{x^{6} \cdot 27}$

Step 9

Simplify the expression.

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Step 9.1

Multiply $2$ by $1$ .

$\frac{2}{x^{6} \cdot 27}$

Step 9.2

Move $27$ to the left of $x^{6}$ .

$\frac{2}{27 x^{6}}$