Pre-Algebra Examples

$\frac{\sqrt[5]{20 x^{7}}}{\sqrt{5 x}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $20 x^{7}$ as $x^{5} \cdot (20 x^{2})$ .

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Step 1.1.1

Factor out $x^{5}$ .

$\frac{\sqrt[5]{20 (x^{5} x^{2})}}{\sqrt{5 x}}$

Step 1.1.2

Reorder $20$ and $x^{5}$ .

$\frac{\sqrt[5]{x^{5} \cdot 20 x^{2}}}{\sqrt{5 x}}$

Step 1.1.3

Add parentheses.

$\frac{\sqrt[5]{x^{5} \cdot (20 x^{2})}}{\sqrt{5 x}}$

Step 1.2

Pull terms out from under the radical.

$\frac{x \sqrt[5]{20 x^{2}}}{\sqrt{5 x}}$

Step 2

Multiply $\frac{x \sqrt[5]{20 x^{2}}}{\sqrt{5 x}}$ by $\frac{\sqrt{5 x}}{\sqrt{5 x}}$ .

$\frac{x \sqrt[5]{20 x^{2}}}{\sqrt{5 x}} \cdot \frac{\sqrt{5 x}}{\sqrt{5 x}}$

Step 3

Simplify terms.

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Step 3.1

Combine and simplify the denominator.

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Step 3.1.1

Multiply $\frac{x \sqrt[5]{20 x^{2}}}{\sqrt{5 x}}$ by $\frac{\sqrt{5 x}}{\sqrt{5 x}}$ .

$\frac{x \sqrt[5]{20 x^{2}} \sqrt{5 x}}{\sqrt{5 x} \sqrt{5 x}}$

Step 3.1.2

Raise $\sqrt{5 x}$ to the power of $1$ .

$\frac{x \sqrt[5]{20 x^{2}} \sqrt{5 x}}{{\sqrt{5 x}}^{1} \sqrt{5 x}}$

Step 3.1.3

Raise $\sqrt{5 x}$ to the power of $1$ .

$\frac{x \sqrt[5]{20 x^{2}} \sqrt{5 x}}{{\sqrt{5 x}}^{1} {\sqrt{5 x}}^{1}}$

Step 3.1.4

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

$\frac{x \sqrt[5]{20 x^{2}} \sqrt{5 x}}{{\sqrt{5 x}}^{1 + 1}}$

Step 3.1.5

Add $1$ and $1$ .

$\frac{x \sqrt[5]{20 x^{2}} \sqrt{5 x}}{{\sqrt{5 x}}^{2}}$

Step 3.1.6

Rewrite ${\sqrt{5 x}}^{2}$ as $5 x$ .

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Step 3.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x}$ as ${(5 x)}^{\frac{1}{2}}$ .

$\frac{x \sqrt[5]{20 x^{2}} \sqrt{5 x}}{{({(5 x)}^{\frac{1}{2}})}^{2}}$

Step 3.1.6.2

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

$\frac{x \sqrt[5]{20 x^{2}} \sqrt{5 x}}{{(5 x)}^{\frac{1}{2} \cdot 2}}$

Step 3.1.6.3

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

$\frac{x \sqrt[5]{20 x^{2}} \sqrt{5 x}}{{(5 x)}^{\frac{2}{2}}}$

Step 3.1.6.4

Cancel the common factor of $2$ .

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Step 3.1.6.4.1

Cancel the common factor.

$\frac{x \sqrt[5]{20 x^{2}} \sqrt{5 x}}{{(5 x)}^{\frac{2}{2}}}$

Step 3.1.6.4.2

Rewrite the expression.

$\frac{x \sqrt[5]{20 x^{2}} \sqrt{5 x}}{{(5 x)}^{1}}$

Step 3.1.6.5

Simplify.

$\frac{x \sqrt[5]{20 x^{2}} \sqrt{5 x}}{5 x}$

Step 3.2

Cancel the common factor of $x$ .

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Step 3.2.1

Cancel the common factor.

$\frac{x \sqrt[5]{20 x^{2}} \sqrt{5 x}}{5 x}$

Step 3.2.2

Rewrite the expression.

$\frac{\sqrt[5]{20 x^{2}} \sqrt{5 x}}{5}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite the expression using the least common index of $10$ .

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Step 4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{20 x^{2}}$ as ${(20 x^{2})}^{\frac{1}{5}}$ .

$\frac{{(20 x^{2})}^{\frac{1}{5}} \sqrt{5 x}}{5}$

Step 4.1.2

Rewrite ${(20 x^{2})}^{\frac{1}{5}}$ as ${(20 x^{2})}^{\frac{2}{10}}$ .

$\frac{{(20 x^{2})}^{\frac{2}{10}} \sqrt{5 x}}{5}$

Step 4.1.3

Rewrite ${(20 x^{2})}^{\frac{2}{10}}$ as $\sqrt[10]{{(20 x^{2})}^{2}}$ .

$\frac{\sqrt[10]{{(20 x^{2})}^{2}} \sqrt{5 x}}{5}$

Step 4.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x}$ as ${(5 x)}^{\frac{1}{2}}$ .

$\frac{\sqrt[10]{{(20 x^{2})}^{2}} {(5 x)}^{\frac{1}{2}}}{5}$

Step 4.1.5

Rewrite ${(5 x)}^{\frac{1}{2}}$ as ${(5 x)}^{\frac{5}{10}}$ .

$\frac{\sqrt[10]{{(20 x^{2})}^{2}} {(5 x)}^{\frac{5}{10}}}{5}$

Step 4.1.6

Rewrite ${(5 x)}^{\frac{5}{10}}$ as $\sqrt[10]{{(5 x)}^{5}}$ .

$\frac{\sqrt[10]{{(20 x^{2})}^{2}} \sqrt[10]{{(5 x)}^{5}}}{5}$

Step 4.2

Combine using the product rule for radicals.

$\frac{\sqrt[10]{{(20 x^{2})}^{2} {(5 x)}^{5}}}{5}$

Step 4.3

Apply the product rule to $20 x^{2}$ .

$\frac{\sqrt[10]{20^{2} {(x^{2})}^{2} {(5 x)}^{5}}}{5}$

Step 4.4

Apply the product rule to $5 x$ .

$\frac{\sqrt[10]{20^{2} {(x^{2})}^{2} \cdot 5^{5} x^{5}}}{5}$

Step 4.5

Raise $20$ to the power of $2$ .

$\frac{\sqrt[10]{400 {(x^{2})}^{2} \cdot 5^{5} x^{5}}}{5}$

Step 4.6

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

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Step 4.6.1

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

$\frac{\sqrt[10]{400 x^{2 \cdot 2} \cdot 5^{5} x^{5}}}{5}$

Step 4.6.2

Multiply $2$ by $2$ .

$\frac{\sqrt[10]{400 x^{4} \cdot 5^{5} x^{5}}}{5}$

Step 4.7

Raise $5$ to the power of $5$ .

$\frac{\sqrt[10]{400 x^{4} \cdot 3125 x^{5}}}{5}$

Step 4.8

Combine exponents.

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Step 4.8.1

Multiply $3125$ by $400$ .

$\frac{\sqrt[10]{1250000 x^{4} x^{5}}}{5}$

Step 4.8.2

Multiply $x^{4}$ by $x^{5}$ by adding the exponents.

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Step 4.8.2.1

Move $x^{5}$ .

$\frac{\sqrt[10]{1250000 (x^{5} x^{4})}}{5}$

Step 4.8.2.2

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

$\frac{\sqrt[10]{1250000 x^{5 + 4}}}{5}$

Step 4.8.2.3

Add $5$ and $4$ .

$\frac{\sqrt[10]{1250000 x^{9}}}{5}$