Pre-Algebra Examples

$\frac{\sqrt{40 x^{5}}}{\sqrt{5 x^{- 7}}}$

Step 1

Combine $\sqrt{40 x^{5}}$ and $\sqrt{5 x^{- 7}}$ into a single radical.

$\sqrt{\frac{40 x^{5}}{5 x^{- 7}}}$

Step 2

Reduce the expression $\frac{40 x^{5}}{5 x^{- 7}}$ by cancelling the common factors.

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Step 2.1

Factor $5$ out of $40 x^{5}$ .

$\sqrt{\frac{5 (8 x^{5})}{5 x^{- 7}}}$

Step 2.2

Factor $5$ out of $5 x^{- 7}$ .

$\sqrt{\frac{5 (8 x^{5})}{5 (x^{- 7})}}$

Step 2.3

Cancel the common factor.

$\sqrt{\frac{5 (8 x^{5})}{5 x^{- 7}}}$

Step 2.4

Rewrite the expression.

$\sqrt{\frac{8 x^{5}}{x^{- 7}}}$

Step 3

Move $x^{- 7}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\sqrt{8 x^{5} x^{7}}$

Step 4

Multiply $x^{5}$ by $x^{7}$ by adding the exponents.

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Step 4.1

Move $x^{7}$ .

$\sqrt{8 (x^{7} x^{5})}$

Step 4.2

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

$\sqrt{8 x^{7 + 5}}$

Step 4.3

Add $7$ and $5$ .

$\sqrt{8 x^{12}}$

Step 5

Rewrite $8 x^{12}$ as ${(2 x^{6})}^{2} \cdot 2$ .

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Step 5.1

Factor $4$ out of $8$ .

$\sqrt{4 (2) x^{12}}$

Step 5.2

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

$\sqrt{2^{2} \cdot 2 x^{12}}$

Step 5.3

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

$\sqrt{2^{2} \cdot 2 {(x^{6})}^{2}}$

Step 5.4

Move $2$ .

$\sqrt{2^{2} {(x^{6})}^{2} \cdot 2}$

Step 5.5

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

$\sqrt{{(2 x^{6})}^{2} \cdot 2}$

Step 6

Pull terms out from under the radical.

$2 x^{6} \sqrt{2}$