Trigonometry Examples

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

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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

Step 3

Expand $(2 \sqrt{x} - 5 \sqrt{y}) (\sqrt{x} - \sqrt{y})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $2 \sqrt{x} \sqrt{x}$ .

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Step 4.1.1.1

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

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

Step 4.1.1.2

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

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

Step 4.1.1.3

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

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

Step 4.1.1.4

Add $1$ and $1$ .

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

Step 4.1.2

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

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Step 4.1.2.1

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

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

Step 4.1.2.2

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

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

Step 4.1.2.3

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

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

Step 4.1.2.4

Cancel the common factor of $2$ .

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Step 4.1.2.4.1

Cancel the common factor.

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

Step 4.1.2.4.2

Rewrite the expression.

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

Step 4.1.2.5

Simplify.

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

Step 4.1.3

Multiply $2 \sqrt{x} (- \sqrt{y})$ .

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Step 4.1.3.1

Multiply $- 1$ by $2$ .

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

Step 4.1.3.2

Combine using the product rule for radicals.

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

Step 4.1.4

Combine using the product rule for radicals.

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

Step 4.1.5

Multiply $- 5 \sqrt{y} (- \sqrt{y})$ .

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Step 4.1.5.1

Multiply $- 1$ by $- 5$ .

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

Step 4.1.5.2

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

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

Step 4.1.5.3

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

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

Step 4.1.5.4

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

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

Step 4.1.5.5

Add $1$ and $1$ .

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

Step 4.1.6

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

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Step 4.1.6.1

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

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

Step 4.1.6.2

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

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

Step 4.1.6.3

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

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

Step 4.1.6.4

Cancel the common factor of $2$ .

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Step 4.1.6.4.1

Cancel the common factor.

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

Step 4.1.6.4.2

Rewrite the expression.

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

Step 4.1.6.5

Simplify.

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

Step 4.2

Subtract $5 \sqrt{x y}$ from $- 2 \sqrt{y x}$ .

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Step 4.2.1

Reorder $y$ and $x$ .

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

Step 4.2.2

Subtract $5 \sqrt{x y}$ from $- 2 \sqrt{x y}$ .

$\frac{2 x - 7 \sqrt{x y} + 5 y}{x - y}$