Pre-Algebra Examples

$\frac{\sqrt[3]{125 x^{7} y^{9}}}{\sqrt[4]{81 x^{16} y^{80}}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $125 x^{7} y^{9}$ as ${(5 x^{2} y^{3})}^{3} x$ .

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Step 1.1.1

Rewrite $125$ as $5^{3}$ .

$\frac{\sqrt[3]{5^{3} x^{7} y^{9}}}{\sqrt[4]{81 x^{16} y^{80}}}$

Step 1.1.2

Factor out $x^{6}$ .

$\frac{\sqrt[3]{5^{3} (x^{6} x) y^{9}}}{\sqrt[4]{81 x^{16} y^{80}}}$

Step 1.1.3

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

$\frac{\sqrt[3]{5^{3} ({(x^{2})}^{3} x) y^{9}}}{\sqrt[4]{81 x^{16} y^{80}}}$

Step 1.1.4

Rewrite $y^{9}$ as ${(y^{3})}^{3}$ .

$\frac{\sqrt[3]{5^{3} ({(x^{2})}^{3} x) {(y^{3})}^{3}}}{\sqrt[4]{81 x^{16} y^{80}}}$

Step 1.1.5

Move $x$ .

$\frac{\sqrt[3]{5^{3} ({(x^{2})}^{3}) {(y^{3})}^{3} x}}{\sqrt[4]{81 x^{16} y^{80}}}$

Step 1.1.6

Rewrite $5^{3} ({(x^{2})}^{3}) {(y^{3})}^{3}$ as ${(5 x^{2} y^{3})}^{3}$ .

$\frac{\sqrt[3]{{(5 x^{2} y^{3})}^{3} x}}{\sqrt[4]{81 x^{16} y^{80}}}$

Step 1.2

Pull terms out from under the radical.

$\frac{5 x^{2} y^{3} \sqrt[3]{x}}{\sqrt[4]{81 x^{16} y^{80}}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $81 x^{16} y^{80}$ as ${(3 x^{4} y^{20})}^{4}$ .

$\frac{5 x^{2} y^{3} \sqrt[3]{x}}{\sqrt[4]{{(3 x^{4} y^{20})}^{4}}}$

Step 2.2

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

$\frac{5 x^{2} y^{3} \sqrt[3]{x}}{3 x^{4} y^{20}}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $x^{2}$ and $x^{4}$ .

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Step 3.1.1

Factor $x^{2}$ out of $5 x^{2} y^{3} \sqrt[3]{x}$ .

$\frac{x^{2} (5 y^{3} \sqrt[3]{x})}{3 x^{4} y^{20}}$

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $x^{2}$ out of $3 x^{4} y^{20}$ .

$\frac{x^{2} (5 y^{3} \sqrt[3]{x})}{x^{2} (3 x^{2} y^{20})}$

Step 3.1.2.2

Cancel the common factor.

$\frac{x^{2} (5 y^{3} \sqrt[3]{x})}{x^{2} (3 x^{2} y^{20})}$

Step 3.1.2.3

Rewrite the expression.

$\frac{5 y^{3} \sqrt[3]{x}}{3 x^{2} y^{20}}$

Step 3.2

Cancel the common factor of $y^{3}$ and $y^{20}$ .

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Step 3.2.1

Factor $y^{3}$ out of $5 y^{3} \sqrt[3]{x}$ .

$\frac{y^{3} (5 \sqrt[3]{x})}{3 x^{2} y^{20}}$

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $y^{3}$ out of $3 x^{2} y^{20}$ .

$\frac{y^{3} (5 \sqrt[3]{x})}{y^{3} (3 x^{2} y^{17})}$

Step 3.2.2.2

Cancel the common factor.

$\frac{y^{3} (5 \sqrt[3]{x})}{y^{3} (3 x^{2} y^{17})}$

Step 3.2.2.3

Rewrite the expression.

$\frac{5 \sqrt[3]{x}}{3 x^{2} y^{17}}$