Linear Algebra Examples

$(11 + \sqrt{15}) (11 - \sqrt{15})$

Step 1

Expand $(11 + \sqrt{15}) (11 - \sqrt{15})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$11 (11 - \sqrt{15}) + \sqrt{15} (11 - \sqrt{15})$

Step 1.2

Apply the distributive property.

$11 \cdot 11 + 11 (- \sqrt{15}) + \sqrt{15} (11 - \sqrt{15})$

Step 1.3

Apply the distributive property.

$11 \cdot 11 + 11 (- \sqrt{15}) + \sqrt{15} \cdot 11 + \sqrt{15} (- \sqrt{15})$

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 $11$ by $11$ .

$121 + 11 (- \sqrt{15}) + \sqrt{15} \cdot 11 + \sqrt{15} (- \sqrt{15})$

Step 2.1.2

Multiply $- 1$ by $11$ .

$121 - 11 \sqrt{15} + \sqrt{15} \cdot 11 + \sqrt{15} (- \sqrt{15})$

Step 2.1.3

Move $11$ to the left of $\sqrt{15}$ .

$121 - 11 \sqrt{15} + 11 \cdot \sqrt{15} + \sqrt{15} (- \sqrt{15})$

Step 2.1.4

Multiply $\sqrt{15} (- \sqrt{15})$ .

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Step 2.1.4.1

Raise $\sqrt{15}$ to the power of $1$ .

$121 - 11 \sqrt{15} + 11 \sqrt{15} - ({\sqrt{15}}^{1} \sqrt{15})$

Step 2.1.4.2

Raise $\sqrt{15}$ to the power of $1$ .

$121 - 11 \sqrt{15} + 11 \sqrt{15} - ({\sqrt{15}}^{1} {\sqrt{15}}^{1})$

Step 2.1.4.3

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

$121 - 11 \sqrt{15} + 11 \sqrt{15} - {\sqrt{15}}^{1 + 1}$

Step 2.1.4.4

Add $1$ and $1$ .

$121 - 11 \sqrt{15} + 11 \sqrt{15} - {\sqrt{15}}^{2}$

Step 2.1.5

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 2.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$121 - 11 \sqrt{15} + 11 \sqrt{15} - {(15^{\frac{1}{2}})}^{2}$

Step 2.1.5.2

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

$121 - 11 \sqrt{15} + 11 \sqrt{15} - 15^{\frac{1}{2} \cdot 2}$

Step 2.1.5.3

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

$121 - 11 \sqrt{15} + 11 \sqrt{15} - 15^{\frac{2}{2}}$

Step 2.1.5.4

Cancel the common factor of $2$ .

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Step 2.1.5.4.1

Cancel the common factor.

$121 - 11 \sqrt{15} + 11 \sqrt{15} - 15^{\frac{2}{2}}$

Step 2.1.5.4.2

Rewrite the expression.

$121 - 11 \sqrt{15} + 11 \sqrt{15} - 15^{1}$

Step 2.1.5.5

Evaluate the exponent.

$121 - 11 \sqrt{15} + 11 \sqrt{15} - 1 \cdot 15$

Step 2.1.6

Multiply $- 1$ by $15$ .

$121 - 11 \sqrt{15} + 11 \sqrt{15} - 15$

Step 2.2

Subtract $15$ from $121$ .

$106 - 11 \sqrt{15} + 11 \sqrt{15}$

Step 2.3

Add $- 11 \sqrt{15}$ and $11 \sqrt{15}$ .

$106 + 0$

Step 2.4

Add $106$ and $0$ .

$106$