Trigonometry Examples

$(\sqrt{x} - \sqrt{y}) (\sqrt{x} - \sqrt{y})$

Step 1

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

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Step 1.1

Apply the distributive property.

$\sqrt{x} (\sqrt{x} - \sqrt{y}) - \sqrt{y} (\sqrt{x} - \sqrt{y})$

Step 1.2

Apply the distributive property.

$\sqrt{x} \sqrt{x} + \sqrt{x} (- \sqrt{y}) - \sqrt{y} (\sqrt{x} - \sqrt{y})$

Step 1.3

Apply the distributive property.

$\sqrt{x} \sqrt{x} + \sqrt{x} (- \sqrt{y}) - \sqrt{y} \sqrt{x} - \sqrt{y} (- \sqrt{y})$

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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Step 2.1.1.1

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

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

Step 2.1.1.2

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

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

Step 2.1.1.3

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

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

Step 2.1.1.4

Add $1$ and $1$ .

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

Step 2.1.2

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

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Step 2.1.2.1

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

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

Step 2.1.2.2

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

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

Step 2.1.2.3

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

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

Step 2.1.2.4

Cancel the common factor of $2$ .

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Step 2.1.2.4.1

Cancel the common factor.

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

Step 2.1.2.4.2

Rewrite the expression.

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

Step 2.1.2.5

Simplify.

$x + \sqrt{x} (- \sqrt{y}) - \sqrt{y} \sqrt{x} - \sqrt{y} (- \sqrt{y})$

Step 2.1.3

Rewrite using the commutative property of multiplication.

$x - \sqrt{x} \sqrt{y} - \sqrt{y} \sqrt{x} - \sqrt{y} (- \sqrt{y})$

Step 2.1.4

Combine using the product rule for radicals.

$x - \sqrt{y x} - \sqrt{y} \sqrt{x} - \sqrt{y} (- \sqrt{y})$

Step 2.1.5

Combine using the product rule for radicals.

$x - \sqrt{y x} - \sqrt{x y} - \sqrt{y} (- \sqrt{y})$

Step 2.1.6

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

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Step 2.1.6.1

Multiply $- 1$ by $- 1$ .

$x - \sqrt{y x} - \sqrt{x y} + 1 \sqrt{y} \sqrt{y}$

Step 2.1.6.2

Multiply $\sqrt{y}$ by $1$ .

$x - \sqrt{y x} - \sqrt{x y} + \sqrt{y} \sqrt{y}$

Step 2.1.6.3

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

$x - \sqrt{y x} - \sqrt{x y} + {\sqrt{y}}^{1} \sqrt{y}$

Step 2.1.6.4

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

$x - \sqrt{y x} - \sqrt{x y} + {\sqrt{y}}^{1} {\sqrt{y}}^{1}$

Step 2.1.6.5

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

$x - \sqrt{y x} - \sqrt{x y} + {\sqrt{y}}^{1 + 1}$

Step 2.1.6.6

Add $1$ and $1$ .

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

Step 2.1.7

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

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Step 2.1.7.1

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

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

Step 2.1.7.2

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

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

Step 2.1.7.3

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

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

Step 2.1.7.4

Cancel the common factor of $2$ .

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Step 2.1.7.4.1

Cancel the common factor.

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

Step 2.1.7.4.2

Rewrite the expression.

$x - \sqrt{y x} - \sqrt{x y} + y^{1}$

Step 2.1.7.5

Simplify.

$x - \sqrt{y x} - \sqrt{x y} + y$

Step 2.2

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

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Step 2.2.1

Reorder $y$ and $x$ .

$x - \sqrt{x y} - \sqrt{x y} + y$

Step 2.2.2

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

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

Step 3

Move $- 2 \sqrt{x y}$ .

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