Finite Math Examples

$\frac{256 x^{2} - y^{2}}{\sqrt[4]{x} - \sqrt{y}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $256 x^{2}$ as ${(16 x)}^{2}$ .

$\frac{{(16 x)}^{2} - y^{2}}{\sqrt[4]{x} - \sqrt{y}}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 16 x$ and $b = y$ .

$\frac{(16 x + y) (16 x - y)}{\sqrt[4]{x} - \sqrt{y}}$

Step 2

Multiply $\frac{(16 x + y) (16 x - y)}{\sqrt[4]{x} - \sqrt{y}}$ by $\frac{\sqrt[4]{x} + \sqrt{y}}{\sqrt[4]{x} + \sqrt{y}}$ .

$\frac{(16 x + y) (16 x - y)}{\sqrt[4]{x} - \sqrt{y}} \cdot \frac{\sqrt[4]{x} + \sqrt{y}}{\sqrt[4]{x} + \sqrt{y}}$

Step 3

Multiply $\frac{(16 x + y) (16 x - y)}{\sqrt[4]{x} - \sqrt{y}}$ by $\frac{\sqrt[4]{x} + \sqrt{y}}{\sqrt[4]{x} + \sqrt{y}}$ .

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x} + \sqrt{y})}{(\sqrt[4]{x} - \sqrt{y}) (\sqrt[4]{x} + \sqrt{y})}$

Step 4

Expand the denominator using the FOIL method.

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x} + \sqrt{y})}{{\sqrt[4]{x}}^{2} + \sqrt[4]{x y^{2}} - \sqrt[4]{x y^{2}} - {\sqrt{y}}^{2}}$

Step 5

Simplify.

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Step 5.1

Rewrite ${\sqrt[4]{x}}^{2}$ as $\sqrt{x}$ .

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Step 5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x} + \sqrt{y})}{{(x^{\frac{1}{4}})}^{2} - {\sqrt{y}}^{2}}$

Step 5.1.2

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

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x} + \sqrt{y})}{x^{\frac{1}{4} \cdot 2} - {\sqrt{y}}^{2}}$

Step 5.1.3

Combine $\frac{1}{4}$ and $2$ .

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x} + \sqrt{y})}{x^{\frac{2}{4}} - {\sqrt{y}}^{2}}$

Step 5.1.4

Cancel the common factor of $2$ and $4$ .

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Step 5.1.4.1

Factor $2$ out of $2$ .

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x} + \sqrt{y})}{x^{\frac{2 (1)}{4}} - {\sqrt{y}}^{2}}$

Step 5.1.4.2

Cancel the common factors.

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Step 5.1.4.2.1

Factor $2$ out of $4$ .

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x} + \sqrt{y})}{x^{\frac{2 \cdot 1}{2 \cdot 2}} - {\sqrt{y}}^{2}}$

Step 5.1.4.2.2

Cancel the common factor.

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x} + \sqrt{y})}{x^{\frac{2 \cdot 1}{2 \cdot 2}} - {\sqrt{y}}^{2}}$

Step 5.1.4.2.3

Rewrite the expression.

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x} + \sqrt{y})}{x^{\frac{1}{2}} - {\sqrt{y}}^{2}}$

Step 5.1.5

Rewrite $x^{\frac{1}{2}}$ as $\sqrt{x}$ .

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x} + \sqrt{y})}{\sqrt{x} - {\sqrt{y}}^{2}}$

Step 5.2

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

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Step 5.2.1

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

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x} + \sqrt{y})}{\sqrt{x} - {(y^{\frac{1}{2}})}^{2}}$

Step 5.2.2

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

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x} + \sqrt{y})}{\sqrt{x} - y^{\frac{1}{2} \cdot 2}}$

Step 5.2.3

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

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x} + \sqrt{y})}{\sqrt{x} - y^{\frac{2}{2}}}$

Step 5.2.4

Cancel the common factor of $2$ .

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Step 5.2.4.1

Cancel the common factor.

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x} + \sqrt{y})}{\sqrt{x} - y^{\frac{2}{2}}}$

Step 5.2.4.2

Rewrite the expression.

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x} + \sqrt{y})}{\sqrt{x} - y^{1}}$

Step 5.2.5

Simplify.

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x} + \sqrt{y})}{\sqrt{x} - y}$

Step 6

Multiply $\frac{(16 x + y) (16 x - y) (\sqrt[4]{x} + \sqrt{y})}{\sqrt{x} - y}$ by $\frac{\sqrt{x} + y}{\sqrt{x} + y}$ .

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x} + \sqrt{y})}{\sqrt{x} - y} \cdot \frac{\sqrt{x} + y}{\sqrt{x} + y}$

Step 7

Combine fractions.

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Step 7.1

Multiply $\frac{(16 x + y) (16 x - y) (\sqrt[4]{x} + \sqrt{y})}{\sqrt{x} - y}$ by $\frac{\sqrt{x} + y}{\sqrt{x} + y}$ .

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x} + \sqrt{y}) (\sqrt{x} + y)}{(\sqrt{x} - y) (\sqrt{x} + y)}$

Step 7.2

Expand the denominator using the FOIL method.

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x} + \sqrt{y}) (\sqrt{x} + y)}{{\sqrt{x}}^{2} + \sqrt{x} y - y \sqrt{x} - y^{2}}$

Step 7.3

Simplify.

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Step 7.3.1

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

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x} + \sqrt{y}) (\sqrt{x} + y)}{{(x^{\frac{1}{2}})}^{2} - y^{2}}$

Step 7.3.2

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

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x} + \sqrt{y}) (\sqrt{x} + y)}{x^{\frac{1}{2} \cdot 2} - y^{2}}$

Step 7.3.3

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

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x} + \sqrt{y}) (\sqrt{x} + y)}{x^{\frac{2}{2}} - y^{2}}$

Step 7.3.4

Cancel the common factor of $2$ .

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Step 7.3.4.1

Cancel the common factor.

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x} + \sqrt{y}) (\sqrt{x} + y)}{x^{\frac{2}{2}} - y^{2}}$

Step 7.3.4.2

Rewrite the expression.

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x} + \sqrt{y}) (\sqrt{x} + y)}{x^{1} - y^{2}}$

Step 7.3.5

Simplify.

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x} + \sqrt{y}) (\sqrt{x} + y)}{x - y^{2}}$

Step 7.4

Group $\sqrt{x} + y$ and $\sqrt[4]{x} + \sqrt{y}$ together.

$\frac{(16 x + y) (16 x - y) ((\sqrt{x} + y) (\sqrt[4]{x} + \sqrt{y}))}{x - y^{2}}$

Step 8

Expand $(\sqrt{x} + y) (\sqrt[4]{x} + \sqrt{y})$ using the FOIL Method.

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Step 8.1

Apply the distributive property.

$\frac{(16 x + y) (16 x - y) (\sqrt{x} (\sqrt[4]{x} + \sqrt{y}) + y (\sqrt[4]{x} + \sqrt{y}))}{x - y^{2}}$

Step 8.2

Apply the distributive property.

$\frac{(16 x + y) (16 x - y) (\sqrt{x} \sqrt[4]{x} + \sqrt{x} \sqrt{y} + y (\sqrt[4]{x} + \sqrt{y}))}{x - y^{2}}$

Step 8.3

Apply the distributive property.

$\frac{(16 x + y) (16 x - y) (\sqrt{x} \sqrt[4]{x} + \sqrt{x} \sqrt{y} + y \sqrt[4]{x} + y \sqrt{y})}{x - y^{2}}$

Step 9

Simplify each term.

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Step 9.1

Multiply $\sqrt{x} \sqrt[4]{x}$ .

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Step 9.1.1

Rewrite the expression using the least common index of $4$ .

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Step 9.1.1.1

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

$\frac{(16 x + y) (16 x - y) (x^{\frac{1}{2}} \sqrt[4]{x} + \sqrt{x} \sqrt{y} + y \sqrt[4]{x} + y \sqrt{y})}{x - y^{2}}$

Step 9.1.1.2

Rewrite $x^{\frac{1}{2}}$ as $x^{\frac{2}{4}}$ .

$\frac{(16 x + y) (16 x - y) (x^{\frac{2}{4}} \sqrt[4]{x} + \sqrt{x} \sqrt{y} + y \sqrt[4]{x} + y \sqrt{y})}{x - y^{2}}$

Step 9.1.1.3

Rewrite $x^{\frac{2}{4}}$ as $\sqrt[4]{x^{2}}$ .

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x^{2}} \sqrt[4]{x} + \sqrt{x} \sqrt{y} + y \sqrt[4]{x} + y \sqrt{y})}{x - y^{2}}$

Step 9.1.2

Combine using the product rule for radicals.

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x^{2} x} + \sqrt{x} \sqrt{y} + y \sqrt[4]{x} + y \sqrt{y})}{x - y^{2}}$

Step 9.1.3

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 9.1.3.1

Multiply $x^{2}$ by $x$ .

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Step 9.1.3.1.1

Raise $x$ to the power of $1$ .

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x^{2} x^{1}} + \sqrt{x} \sqrt{y} + y \sqrt[4]{x} + y \sqrt{y})}{x - y^{2}}$

Step 9.1.3.1.2

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

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x^{2 + 1}} + \sqrt{x} \sqrt{y} + y \sqrt[4]{x} + y \sqrt{y})}{x - y^{2}}$

Step 9.1.3.2

Add $2$ and $1$ .

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x^{3}} + \sqrt{x} \sqrt{y} + y \sqrt[4]{x} + y \sqrt{y})}{x - y^{2}}$

Step 9.2

Combine using the product rule for radicals.

$\frac{(16 x + y) (16 x - y) (\sqrt[4]{x^{3}} + \sqrt{x y} + y \sqrt[4]{x} + y \sqrt{y})}{x - y^{2}}$

Step 10

Simplify the numerator.

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Step 10.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x^{3}}$ as $x^{\frac{3}{4}}$ .

$\frac{(16 x + y) (16 x - y) (x^{\frac{3}{4}} + \sqrt{x y} + y \sqrt[4]{x} + y \sqrt{y})}{x - y^{2}}$

Step 10.2

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

$\frac{(16 x + y) (16 x - y) (x^{\frac{3}{4}} + {(x y)}^{\frac{1}{2}} + y \sqrt[4]{x} + y \sqrt{y})}{x - y^{2}}$

Step 10.3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

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

Step 10.4

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

$\frac{(16 x + y) (16 x - y) (x^{\frac{3}{4}} + {(x y)}^{\frac{1}{2}} + y x^{\frac{1}{4}} + y \cdot y^{\frac{1}{2}})}{x - y^{2}}$

Step 10.5

Apply the product rule to $x y$ .

$\frac{(16 x + y) (16 x - y) (x^{\frac{3}{4}} + x^{\frac{1}{2}} y^{\frac{1}{2}} + y x^{\frac{1}{4}} + y \cdot y^{\frac{1}{2}})}{x - y^{2}}$

Step 10.6

Multiply $y$ by $y^{\frac{1}{2}}$ by adding the exponents.

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Step 10.6.1

Multiply $y$ by $y^{\frac{1}{2}}$ .

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Step 10.6.1.1

Raise $y$ to the power of $1$ .

$\frac{(16 x + y) (16 x - y) (x^{\frac{3}{4}} + x^{\frac{1}{2}} y^{\frac{1}{2}} + y x^{\frac{1}{4}} + y^{1} y^{\frac{1}{2}})}{x - y^{2}}$

Step 10.6.1.2

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

$\frac{(16 x + y) (16 x - y) (x^{\frac{3}{4}} + x^{\frac{1}{2}} y^{\frac{1}{2}} + y x^{\frac{1}{4}} + y^{1 + \frac{1}{2}})}{x - y^{2}}$

Step 10.6.2

Write $1$ as a fraction with a common denominator.

$\frac{(16 x + y) (16 x - y) (x^{\frac{3}{4}} + x^{\frac{1}{2}} y^{\frac{1}{2}} + y x^{\frac{1}{4}} + y^{\frac{2}{2} + \frac{1}{2}})}{x - y^{2}}$

Step 10.6.3

Combine the numerators over the common denominator.

$\frac{(16 x + y) (16 x - y) (x^{\frac{3}{4}} + x^{\frac{1}{2}} y^{\frac{1}{2}} + y x^{\frac{1}{4}} + y^{\frac{2 + 1}{2}})}{x - y^{2}}$

Step 10.6.4

Add $2$ and $1$ .

$\frac{(16 x + y) (16 x - y) (x^{\frac{3}{4}} + x^{\frac{1}{2}} y^{\frac{1}{2}} + y x^{\frac{1}{4}} + y^{\frac{3}{2}})}{x - y^{2}}$

Step 10.7

Rewrite $x^{\frac{3}{4}} + x^{\frac{1}{2}} y^{\frac{1}{2}} + y x^{\frac{1}{4}} + y^{\frac{3}{2}}$ in a factored form.

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Step 10.7.1

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{3}{4}} + x^{\frac{1}{2}} y^{\frac{1}{2}}$ .

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Step 10.7.1.1

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{3}{4}}$ .

$\frac{(16 x + y) (16 x - y) (x^{\frac{1}{2}} x^{\frac{1}{4}} + x^{\frac{1}{2}} y^{\frac{1}{2}} + y x^{\frac{1}{4}} + y^{\frac{3}{2}})}{x - y^{2}}$

Step 10.7.1.2

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} y^{\frac{1}{2}}$ .

$\frac{(16 x + y) (16 x - y) (x^{\frac{1}{2}} x^{\frac{1}{4}} + x^{\frac{1}{2}} (y^{\frac{1}{2}}) + y x^{\frac{1}{4}} + y^{\frac{3}{2}})}{x - y^{2}}$

Step 10.7.1.3

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} x^{\frac{1}{4}} + x^{\frac{1}{2}} (y^{\frac{1}{2}})$ .

$\frac{(16 x + y) (16 x - y) (x^{\frac{1}{2}} (x^{\frac{1}{4}} + y^{\frac{1}{2}}) + y x^{\frac{1}{4}} + y^{\frac{3}{2}})}{x - y^{2}}$

Step 10.7.2

Factor $y$ out of $y x^{\frac{1}{4}} + y^{\frac{3}{2}}$ .

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Step 10.7.2.1

Factor $y$ out of $y x^{\frac{1}{4}}$ .

$\frac{(16 x + y) (16 x - y) (x^{\frac{1}{2}} (x^{\frac{1}{4}} + y^{\frac{1}{2}}) + y (x^{\frac{1}{4}}) + y^{\frac{3}{2}})}{x - y^{2}}$

Step 10.7.2.2

Factor $y$ out of $y^{\frac{3}{2}}$ .

$\frac{(16 x + y) (16 x - y) (x^{\frac{1}{2}} (x^{\frac{1}{4}} + y^{\frac{1}{2}}) + y (x^{\frac{1}{4}}) + y \cdot y^{\frac{1}{2}})}{x - y^{2}}$

Step 10.7.2.3

Factor $y$ out of $y (x^{\frac{1}{4}}) + y \cdot y^{\frac{1}{2}}$ .

$\frac{(16 x + y) (16 x - y) (x^{\frac{1}{2}} (x^{\frac{1}{4}} + y^{\frac{1}{2}}) + y (x^{\frac{1}{4}} + y^{\frac{1}{2}}))}{x - y^{2}}$

Step 10.7.3

Factor $x^{\frac{1}{4}} + y^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} (x^{\frac{1}{4}} + y^{\frac{1}{2}}) + y (x^{\frac{1}{4}} + y^{\frac{1}{2}})$ .

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Step 10.7.3.1

Factor $x^{\frac{1}{4}} + y^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} (x^{\frac{1}{4}} + y^{\frac{1}{2}})$ .

$\frac{(16 x + y) (16 x - y) ((x^{\frac{1}{4}} + y^{\frac{1}{2}}) x^{\frac{1}{2}} + y (x^{\frac{1}{4}} + y^{\frac{1}{2}}))}{x - y^{2}}$

Step 10.7.3.2

Factor $x^{\frac{1}{4}} + y^{\frac{1}{2}}$ out of $y (x^{\frac{1}{4}} + y^{\frac{1}{2}})$ .

$\frac{(16 x + y) (16 x - y) ((x^{\frac{1}{4}} + y^{\frac{1}{2}}) x^{\frac{1}{2}} + (x^{\frac{1}{4}} + y^{\frac{1}{2}}) y)}{x - y^{2}}$

Step 10.7.3.3

Factor $x^{\frac{1}{4}} + y^{\frac{1}{2}}$ out of $(x^{\frac{1}{4}} + y^{\frac{1}{2}}) x^{\frac{1}{2}} + (x^{\frac{1}{4}} + y^{\frac{1}{2}}) y$ .

$\frac{(16 x + y) (16 x - y) ((x^{\frac{1}{4}} + y^{\frac{1}{2}}) (x^{\frac{1}{2}} + y))}{x - y^{2}}$

$\frac{(16 x + y) (16 x - y) (x^{\frac{1}{4}} + y^{\frac{1}{2}}) (x^{\frac{1}{2}} + y)}{x - y^{2}}$