Precalculus Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $\sqrt{y} \sqrt{y} + \sqrt{y} (- \sqrt{2}) + \sqrt{2} \sqrt{y} + \sqrt{2} (- \sqrt{2})$ .

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Step 2.1.1

Reorder the factors in the terms $\sqrt{y} (- \sqrt{2})$ and $\sqrt{2} \sqrt{y}$ .

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

Step 2.1.2

Add $- \sqrt{2} \sqrt{y}$ and $\sqrt{2} \sqrt{y}$ .

$\sqrt{y} \sqrt{y} + 0 + \sqrt{2} (- \sqrt{2})$

Step 2.1.3

Add $\sqrt{y} \sqrt{y}$ and $0$ .

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

Step 2.2

Simplify each term.

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.1.4

Add $1$ and $1$ .

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

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

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

Step 2.2.2.4

Cancel the common factor of $2$ .

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Step 2.2.2.4.1

Cancel the common factor.

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

Step 2.2.2.4.2

Rewrite the expression.

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

Step 2.2.2.5

Simplify.

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

Step 2.2.3

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

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Step 2.2.3.1

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 2.2.3.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 2.2.3.3

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

$y - {\sqrt{2}}^{1 + 1}$

Step 2.2.3.4

Add $1$ and $1$ .

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

Step 2.2.4

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

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Step 2.2.4.1

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

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

Step 2.2.4.2

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

$y - 2^{\frac{1}{2} \cdot 2}$

Step 2.2.4.3

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

$y - 2^{\frac{2}{2}}$

Step 2.2.4.4

Cancel the common factor of $2$ .

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Step 2.2.4.4.1

Cancel the common factor.

$y - 2^{\frac{2}{2}}$

Step 2.2.4.4.2

Rewrite the expression.

$y - 2^{1}$

Step 2.2.4.5

Evaluate the exponent.

$y - 1 \cdot 2$

Step 2.2.5

Multiply $- 1$ by $2$ .

$y - 2$