Precalculus Examples

${(8 + \sqrt{- 5})}^{2}$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite $- 5$ as $- 1 (5)$ .

${(8 + \sqrt{- 1 (5)})}^{2}$

Step 1.1.2

Rewrite $\sqrt{- 1 (5)}$ as $\sqrt{- 1} \cdot \sqrt{5}$ .

${(8 + \sqrt{- 1} \cdot \sqrt{5})}^{2}$

Step 1.1.3

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

${(8 + i \sqrt{5})}^{2}$

Step 1.2

Rewrite ${(8 + i \sqrt{5})}^{2}$ as $(8 + i \sqrt{5}) (8 + i \sqrt{5})$ .

$(8 + i \sqrt{5}) (8 + i \sqrt{5})$

Step 2

Expand $(8 + i \sqrt{5}) (8 + i \sqrt{5})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$8 (8 + i \sqrt{5}) + i \sqrt{5} (8 + i \sqrt{5})$

Step 2.2

Apply the distributive property.

$8 \cdot 8 + 8 (i \sqrt{5}) + i \sqrt{5} (8 + i \sqrt{5})$

Step 2.3

Apply the distributive property.

$8 \cdot 8 + 8 (i \sqrt{5}) + i \sqrt{5} \cdot 8 + i \sqrt{5} (i \sqrt{5})$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $8$ by $8$ .

$64 + 8 (i \sqrt{5}) + i \sqrt{5} \cdot 8 + i \sqrt{5} (i \sqrt{5})$

Step 3.1.2

Move $8$ to the left of $i \sqrt{5}$ .

$64 + 8 i \sqrt{5} + 8 \cdot (i \sqrt{5}) + i \sqrt{5} (i \sqrt{5})$

Step 3.1.3

Multiply $i \sqrt{5} (i \sqrt{5})$ .

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Step 3.1.3.1

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

$64 + 8 i \sqrt{5} + 8 i \sqrt{5} + i^{1} i \sqrt{5} \sqrt{5}$

Step 3.1.3.2

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

$64 + 8 i \sqrt{5} + 8 i \sqrt{5} + i^{1} i^{1} \sqrt{5} \sqrt{5}$

Step 3.1.3.3

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

$64 + 8 i \sqrt{5} + 8 i \sqrt{5} + i^{1 + 1} \sqrt{5} \sqrt{5}$

Step 3.1.3.4

Add $1$ and $1$ .

$64 + 8 i \sqrt{5} + 8 i \sqrt{5} + i^{2} \sqrt{5} \sqrt{5}$

Step 3.1.3.5

Raise $\sqrt{5}$ to the power of $1$ .

$64 + 8 i \sqrt{5} + 8 i \sqrt{5} + i^{2} ({\sqrt{5}}^{1} \sqrt{5})$

Step 3.1.3.6

Raise $\sqrt{5}$ to the power of $1$ .

$64 + 8 i \sqrt{5} + 8 i \sqrt{5} + i^{2} ({\sqrt{5}}^{1} {\sqrt{5}}^{1})$

Step 3.1.3.7

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

$64 + 8 i \sqrt{5} + 8 i \sqrt{5} + i^{2} {\sqrt{5}}^{1 + 1}$

Step 3.1.3.8

Add $1$ and $1$ .

$64 + 8 i \sqrt{5} + 8 i \sqrt{5} + i^{2} {\sqrt{5}}^{2}$

Step 3.1.4

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

$64 + 8 i \sqrt{5} + 8 i \sqrt{5} - 1 {\sqrt{5}}^{2}$

Step 3.1.5

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$64 + 8 i \sqrt{5} + 8 i \sqrt{5} - 1 {(5^{\frac{1}{2}})}^{2}$

Step 3.1.5.2

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

$64 + 8 i \sqrt{5} + 8 i \sqrt{5} - 1 \cdot 5^{\frac{1}{2} \cdot 2}$

Step 3.1.5.3

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

$64 + 8 i \sqrt{5} + 8 i \sqrt{5} - 1 \cdot 5^{\frac{2}{2}}$

Step 3.1.5.4

Cancel the common factor of $2$ .

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Step 3.1.5.4.1

Cancel the common factor.

$64 + 8 i \sqrt{5} + 8 i \sqrt{5} - 1 \cdot 5^{\frac{2}{2}}$

Step 3.1.5.4.2

Rewrite the expression.

$64 + 8 i \sqrt{5} + 8 i \sqrt{5} - 1 \cdot 5^{1}$

Step 3.1.5.5

Evaluate the exponent.

$64 + 8 i \sqrt{5} + 8 i \sqrt{5} - 1 \cdot 5$

Step 3.1.6

Multiply $- 1$ by $5$ .

$64 + 8 i \sqrt{5} + 8 i \sqrt{5} - 5$

Step 3.2

Subtract $5$ from $64$ .

$8 i \sqrt{5} + 8 i \sqrt{5} + 59$

Step 3.3

Add $8 i \sqrt{5}$ and $8 i \sqrt{5}$ .

$16 i \sqrt{5} + 59$

Step 4

Reorder $16 i \sqrt{5}$ and $59$ .

$59 + 16 i \sqrt{5}$