Basic Math Examples

$(3 + \sqrt{- 16}) (3 + \sqrt{- 64})$

Step 1

Rewrite $- 16$ as $- 1 (16)$ .

$(3 + \sqrt{- 1 (16)}) (3 + \sqrt{- 64})$

Step 2

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$(3 + \sqrt{- 1} \cdot \sqrt{16}) (3 + \sqrt{- 64})$

Step 3

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

$(3 + i \cdot \sqrt{16}) (3 + \sqrt{- 64})$

Step 4

Rewrite $- 64$ as $- 1 (64)$ .

$(3 + i \cdot \sqrt{16}) (3 + \sqrt{- 1 (64)})$

Step 5

Rewrite $\sqrt{- 1 (64)}$ as $\sqrt{- 1} \cdot \sqrt{64}$ .

$(3 + i \cdot \sqrt{16}) (3 + \sqrt{- 1} \cdot \sqrt{64})$

Step 6

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

$(3 + i \cdot \sqrt{16}) (3 + i \cdot \sqrt{64})$

Step 7

Simplify terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

Rewrite $16$ as $4^{2}$ .

$(3 + i \cdot \sqrt{4^{2}}) (3 + i \cdot \sqrt{64})$

Step 7.1.2

Pull terms out from under the radical, assuming positive real numbers.

$(3 + i \cdot 4) (3 + i \cdot \sqrt{64})$

Step 7.1.3

Move $4$ to the left of $i$ .

$(3 + 4 i) (3 + i \cdot \sqrt{64})$

Step 7.2

Simplify each term.

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Step 7.2.1

Rewrite $64$ as $8^{2}$ .

$(3 + 4 i) (3 + i \cdot \sqrt{8^{2}})$

Step 7.2.2

Pull terms out from under the radical, assuming positive real numbers.

$(3 + 4 i) (3 + i \cdot 8)$

Step 7.2.3

Move $8$ to the left of $i$ .

$(3 + 4 i) (3 + 8 i)$

Step 8

Expand $(3 + 4 i) (3 + 8 i)$ using the FOIL Method.

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Step 8.1

Apply the distributive property.

$3 (3 + 8 i) + 4 i (3 + 8 i)$

Step 8.2

Apply the distributive property.

$3 \cdot 3 + 3 (8 i) + 4 i (3 + 8 i)$

Step 8.3

Apply the distributive property.

$3 \cdot 3 + 3 (8 i) + 4 i \cdot 3 + 4 i (8 i)$

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 $3$ by $3$ .

$9 + 3 (8 i) + 4 i \cdot 3 + 4 i (8 i)$

Step 9.1.2

Multiply $8$ by $3$ .

$9 + 24 i + 4 i \cdot 3 + 4 i (8 i)$

Step 9.1.3

Multiply $3$ by $4$ .

$9 + 24 i + 12 i + 4 i (8 i)$

Step 9.1.4

Multiply $4 i (8 i)$ .

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Step 9.1.4.1

Multiply $8$ by $4$ .

$9 + 24 i + 12 i + 32 i i$

Step 9.1.4.2

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

$9 + 24 i + 12 i + 32 (i^{1} i)$

Step 9.1.4.3

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

$9 + 24 i + 12 i + 32 (i^{1} i^{1})$

Step 9.1.4.4

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

$9 + 24 i + 12 i + 32 i^{1 + 1}$

Step 9.1.4.5

Add $1$ and $1$ .

$9 + 24 i + 12 i + 32 i^{2}$

Step 9.1.5

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

$9 + 24 i + 12 i + 32 \cdot - 1$

Step 9.1.6

Multiply $32$ by $- 1$ .

$9 + 24 i + 12 i - 32$

Step 9.2

Subtract $32$ from $9$ .

$- 23 + 24 i + 12 i$

Step 9.3

Add $24 i$ and $12 i$ .

$- 23 + 36 i$