Finite Math Examples

$\frac{5 \sqrt{x}}{\sqrt{x} - 2 \sqrt{y}}$

Step 1

Multiply $\frac{5 \sqrt{x}}{\sqrt{x} - 2 \sqrt{y}}$ by $\frac{\sqrt{x} + 2 \sqrt{y}}{\sqrt{x} + 2 \sqrt{y}}$ .

$\frac{5 \sqrt{x}}{\sqrt{x} - 2 \sqrt{y}} \cdot \frac{\sqrt{x} + 2 \sqrt{y}}{\sqrt{x} + 2 \sqrt{y}}$

Step 2

Multiply $\frac{5 \sqrt{x}}{\sqrt{x} - 2 \sqrt{y}}$ by $\frac{\sqrt{x} + 2 \sqrt{y}}{\sqrt{x} + 2 \sqrt{y}}$ .

$\frac{5 \sqrt{x} (\sqrt{x} + 2 \sqrt{y})}{(\sqrt{x} - 2 \sqrt{y}) (\sqrt{x} + 2 \sqrt{y})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{5 \sqrt{x} (\sqrt{x} + 2 \sqrt{y})}{{\sqrt{x}}^{2} + 2 \sqrt{x y} - 2 \sqrt{x y} - 4 {\sqrt{y}}^{2}}$

Step 4

Simplify.

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Step 4.1

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

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Step 4.1.1

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

$\frac{5 \sqrt{x} (\sqrt{x} + 2 \sqrt{y})}{{(x^{\frac{1}{2}})}^{2} - 4 {\sqrt{y}}^{2}}$

Step 4.1.2

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

$\frac{5 \sqrt{x} (\sqrt{x} + 2 \sqrt{y})}{x^{\frac{1}{2} \cdot 2} - 4 {\sqrt{y}}^{2}}$

Step 4.1.3

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

$\frac{5 \sqrt{x} (\sqrt{x} + 2 \sqrt{y})}{x^{\frac{2}{2}} - 4 {\sqrt{y}}^{2}}$

Step 4.1.4

Cancel the common factor of $2$ .

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Step 4.1.4.1

Cancel the common factor.

$\frac{5 \sqrt{x} (\sqrt{x} + 2 \sqrt{y})}{x^{\frac{2}{2}} - 4 {\sqrt{y}}^{2}}$

Step 4.1.4.2

Rewrite the expression.

$\frac{5 \sqrt{x} (\sqrt{x} + 2 \sqrt{y})}{x^{1} - 4 {\sqrt{y}}^{2}}$

Step 4.1.5

Simplify.

$\frac{5 \sqrt{x} (\sqrt{x} + 2 \sqrt{y})}{x - 4 {\sqrt{y}}^{2}}$

Step 4.2

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

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Step 4.2.1

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

$\frac{5 \sqrt{x} (\sqrt{x} + 2 \sqrt{y})}{x - 4 {(y^{\frac{1}{2}})}^{2}}$

Step 4.2.2

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

$\frac{5 \sqrt{x} (\sqrt{x} + 2 \sqrt{y})}{x - 4 y^{\frac{1}{2} \cdot 2}}$

Step 4.2.3

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

$\frac{5 \sqrt{x} (\sqrt{x} + 2 \sqrt{y})}{x - 4 y^{\frac{2}{2}}}$

Step 4.2.4

Cancel the common factor of $2$ .

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Step 4.2.4.1

Cancel the common factor.

$\frac{5 \sqrt{x} (\sqrt{x} + 2 \sqrt{y})}{x - 4 y^{\frac{2}{2}}}$

Step 4.2.4.2

Rewrite the expression.

$\frac{5 \sqrt{x} (\sqrt{x} + 2 \sqrt{y})}{x - 4 y^{1}}$

Step 4.2.5

Simplify.

$\frac{5 \sqrt{x} (\sqrt{x} + 2 \sqrt{y})}{x - 4 y}$

Step 5

Group $\sqrt{x} + 2 \sqrt{y}$ and $\sqrt{x}$ together.

$\frac{5 ((\sqrt{x} + 2 \sqrt{y}) \sqrt{x})}{x - 4 y}$

Step 6

Apply the distributive property.

$\frac{5 (\sqrt{x} \sqrt{x} + 2 \sqrt{y} \sqrt{x})}{x - 4 y}$

Step 7

Multiply $\sqrt{x} \sqrt{x}$ .

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Step 7.1

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

$\frac{5 ({\sqrt{x}}^{1} \sqrt{x} + 2 \sqrt{y} \sqrt{x})}{x - 4 y}$

Step 7.2

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

$\frac{5 ({\sqrt{x}}^{1} {\sqrt{x}}^{1} + 2 \sqrt{y} \sqrt{x})}{x - 4 y}$

Step 7.3

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

$\frac{5 ({\sqrt{x}}^{1 + 1} + 2 \sqrt{y} \sqrt{x})}{x - 4 y}$

Step 7.4

Add $1$ and $1$ .

$\frac{5 ({\sqrt{x}}^{2} + 2 \sqrt{y} \sqrt{x})}{x - 4 y}$

Step 8

Simplify terms.

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Step 8.1

Combine using the product rule for radicals.

$\frac{5 ({\sqrt{x}}^{2} + 2 \sqrt{x y})}{x - 4 y}$

Step 8.2

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

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Step 8.2.1

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

$\frac{5 ({(x^{\frac{1}{2}})}^{2} + 2 \sqrt{x y})}{x - 4 y}$

Step 8.2.2

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

$\frac{5 (x^{\frac{1}{2} \cdot 2} + 2 \sqrt{x y})}{x - 4 y}$

Step 8.2.3

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

$\frac{5 (x^{\frac{2}{2}} + 2 \sqrt{x y})}{x - 4 y}$

Step 8.2.4

Cancel the common factor of $2$ .

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Step 8.2.4.1

Cancel the common factor.

$\frac{5 (x^{\frac{2}{2}} + 2 \sqrt{x y})}{x - 4 y}$

Step 8.2.4.2

Rewrite the expression.

$\frac{5 (x^{1} + 2 \sqrt{x y})}{x - 4 y}$

Step 8.2.5

Simplify.

$\frac{5 (x + 2 \sqrt{x y})}{x - 4 y}$

Step 9

Simplify the numerator.

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Step 9.1

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

$\frac{5 (x + 2 {(x y)}^{\frac{1}{2}})}{x - 4 y}$

Step 9.2

Apply the product rule to $x y$ .

$\frac{5 (x + 2 (x^{\frac{1}{2}} y^{\frac{1}{2}}))}{x - 4 y}$

Step 9.3

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

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Step 9.3.1

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

$\frac{5 (x^{1} + 2 x^{\frac{1}{2}} y^{\frac{1}{2}})}{x - 4 y}$

Step 9.3.2

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

$\frac{5 (x^{\frac{1}{2}} x^{\frac{1}{2}} + 2 x^{\frac{1}{2}} y^{\frac{1}{2}})}{x - 4 y}$

Step 9.3.3

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

$\frac{5 (x^{\frac{1}{2}} x^{\frac{1}{2}} + x^{\frac{1}{2}} (2 y^{\frac{1}{2}}))}{x - 4 y}$

Step 9.3.4

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

$\frac{5 (x^{\frac{1}{2}} (x^{\frac{1}{2}} + 2 y^{\frac{1}{2}}))}{x - 4 y}$

$\frac{5 x^{\frac{1}{2}} (x^{\frac{1}{2}} + 2 y^{\frac{1}{2}})}{x - 4 y}$