Algebra Examples

$\sqrt{- 64} - 4 \sqrt[3]{- 125} + \sqrt{8}$

Step 1

Simplify each term.

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Step 1.1

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

$\sqrt{- 1 (64)} - 4 \sqrt[3]{- 125} + \sqrt{8}$

Step 1.2

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

$\sqrt{- 1} \cdot \sqrt{64} - 4 \sqrt[3]{- 125} + \sqrt{8}$

Step 1.3

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

$i \cdot \sqrt{64} - 4 \sqrt[3]{- 125} + \sqrt{8}$

Step 1.4

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

$i \cdot \sqrt{8^{2}} - 4 \sqrt[3]{- 125} + \sqrt{8}$

Step 1.5

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

$i \cdot 8 - 4 \sqrt[3]{- 125} + \sqrt{8}$

Step 1.6

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

$8 \cdot i - 4 \sqrt[3]{- 125} + \sqrt{8}$

Step 1.7

Rewrite $- 125$ as ${(- 5)}^{3}$ .

$8 i - 4 \sqrt[3]{{(- 5)}^{3}} + \sqrt{8}$

Step 1.8

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

$8 i - 4 \cdot - 5 + \sqrt{8}$

Step 1.9

Multiply $- 4$ by $- 5$ .

$8 i + 20 + \sqrt{8}$

Step 1.10

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 1.10.1

Factor $4$ out of $8$ .

$8 i + 20 + \sqrt{4 (2)}$

Step 1.10.2

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

$8 i + 20 + \sqrt{2^{2} \cdot 2}$

Step 1.11

Pull terms out from under the radical.

$8 i + 20 + 2 \sqrt{2}$

Step 2

Simplify the expression.

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Step 2.1

Reorder $8 i$ and $20$ .

$20 + 8 i + 2 \sqrt{2}$

Step 2.2

Reorder $20 + 8 i$ and $2 \sqrt{2}$ .

$2 \sqrt{2} + 20 + 8 i$