Precalculus Examples

$(8 - \sqrt{- 11}) (8 + \sqrt{- 11})$

Step 1

Rewrite $- 11$ as $- 1 (11)$ .

$(8 - \sqrt{- 1 (11)}) (8 + \sqrt{- 11})$

Step 2

Rewrite $\sqrt{- 1 (11)}$ as $\sqrt{- 1} \cdot \sqrt{11}$ .

$(8 - (\sqrt{- 1} \cdot \sqrt{11})) (8 + \sqrt{- 11})$

Step 3

Rewrite $\sqrt{- 1}$ as $i$ .

$(8 - (i \cdot \sqrt{11})) (8 + \sqrt{- 11})$

Step 4

Rewrite $- 11$ as $- 1 (11)$ .

$(8 - (i \cdot \sqrt{11})) (8 + \sqrt{- 1 (11)})$

Step 5

Rewrite $\sqrt{- 1 (11)}$ as $\sqrt{- 1} \cdot \sqrt{11}$ .

$(8 - (i \cdot \sqrt{11})) (8 + \sqrt{- 1} \cdot \sqrt{11})$

Step 6

Rewrite $\sqrt{- 1}$ as $i$ .

$(8 - (i \cdot \sqrt{11})) (8 + i \cdot \sqrt{11})$

Step 7

Remove parentheses.

$(8 - i \sqrt{11}) (8 + i \cdot \sqrt{11})$

Step 8

Expand $(8 - i \sqrt{11}) (8 + i \sqrt{11})$ using the FOIL Method.

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Step 8.1

Apply the distributive property.

$8 (8 + i \sqrt{11}) - i \sqrt{11} (8 + i \sqrt{11})$

Step 8.2

Apply the distributive property.

$8 \cdot 8 + 8 (i \sqrt{11}) - i \sqrt{11} (8 + i \sqrt{11})$

Step 8.3

Apply the distributive property.

$8 \cdot 8 + 8 (i \sqrt{11}) - i \sqrt{11} \cdot 8 - i \sqrt{11} (i \sqrt{11})$

Step 9

Simplify and combine like terms.

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Step 9.1

Simplify each term.

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Step 9.1.1

Multiply $8$ by $8$ .

$64 + 8 (i \sqrt{11}) - i \sqrt{11} \cdot 8 - i \sqrt{11} (i \sqrt{11})$

Step 9.1.2

Multiply $8$ by $- 1$ .

$64 + 8 i \sqrt{11} - 8 i \sqrt{11} - i \sqrt{11} (i \sqrt{11})$

Step 9.1.3

Multiply $- i \sqrt{11} (i \sqrt{11})$ .

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Step 9.1.3.1

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

$64 + 8 i \sqrt{11} - 8 i \sqrt{11} - (i^{1} i) \sqrt{11} \sqrt{11}$

Step 9.1.3.2

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

$64 + 8 i \sqrt{11} - 8 i \sqrt{11} - (i^{1} i^{1}) \sqrt{11} \sqrt{11}$

Step 9.1.3.3

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

$64 + 8 i \sqrt{11} - 8 i \sqrt{11} - i^{1 + 1} \sqrt{11} \sqrt{11}$

Step 9.1.3.4

Add $1$ and $1$ .

$64 + 8 i \sqrt{11} - 8 i \sqrt{11} - i^{2} \sqrt{11} \sqrt{11}$

Step 9.1.3.5

Raise $\sqrt{11}$ to the power of $1$ .

$64 + 8 i \sqrt{11} - 8 i \sqrt{11} - i^{2} ({\sqrt{11}}^{1} \sqrt{11})$

Step 9.1.3.6

Raise $\sqrt{11}$ to the power of $1$ .

$64 + 8 i \sqrt{11} - 8 i \sqrt{11} - i^{2} ({\sqrt{11}}^{1} {\sqrt{11}}^{1})$

Step 9.1.3.7

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

$64 + 8 i \sqrt{11} - 8 i \sqrt{11} - i^{2} {\sqrt{11}}^{1 + 1}$

Step 9.1.3.8

Add $1$ and $1$ .

$64 + 8 i \sqrt{11} - 8 i \sqrt{11} - i^{2} {\sqrt{11}}^{2}$

Step 9.1.4

Rewrite $i^{2}$ as $- 1$ .

$64 + 8 i \sqrt{11} - 8 i \sqrt{11} - - 1 {\sqrt{11}}^{2}$

Step 9.1.5

Multiply $- 1$ by $- 1$ .

$64 + 8 i \sqrt{11} - 8 i \sqrt{11} + 1 {\sqrt{11}}^{2}$

Step 9.1.6

Multiply ${\sqrt{11}}^{2}$ by $1$ .

$64 + 8 i \sqrt{11} - 8 i \sqrt{11} + {\sqrt{11}}^{2}$

Step 9.1.7

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 9.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$64 + 8 i \sqrt{11} - 8 i \sqrt{11} + {(11^{\frac{1}{2}})}^{2}$

Step 9.1.7.2

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

$64 + 8 i \sqrt{11} - 8 i \sqrt{11} + 11^{\frac{1}{2} \cdot 2}$

Step 9.1.7.3

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

$64 + 8 i \sqrt{11} - 8 i \sqrt{11} + 11^{\frac{2}{2}}$

Step 9.1.7.4

Cancel the common factor of $2$ .

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Step 9.1.7.4.1

Cancel the common factor.

$64 + 8 i \sqrt{11} - 8 i \sqrt{11} + 11^{\frac{2}{2}}$

Step 9.1.7.4.2

Rewrite the expression.

$64 + 8 i \sqrt{11} - 8 i \sqrt{11} + 11^{1}$

Step 9.1.7.5

Evaluate the exponent.

$64 + 8 i \sqrt{11} - 8 i \sqrt{11} + 11$

Step 9.2

Add $64$ and $11$ .

$8 i \sqrt{11} - 8 i \sqrt{11} + 75$

Step 9.3

Subtract $8 i \sqrt{11}$ from $8 i \sqrt{11}$ .

$0 + 75$

Step 9.4

Add $0$ and $75$ .

$75$