Algebra Examples

$(11 - \sqrt{- 72}) (2 + \sqrt{- 18})$

Step 1

Rewrite $- 72$ as $- 1 (72)$ .

$(11 - \sqrt{- 1 (72)}) (2 + \sqrt{- 18})$

Step 2

Rewrite $\sqrt{- 1 (72)}$ as $\sqrt{- 1} \cdot \sqrt{72}$ .

$(11 - (\sqrt{- 1} \cdot \sqrt{72})) (2 + \sqrt{- 18})$

Step 3

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

$(11 - (i \cdot \sqrt{72})) (2 + \sqrt{- 18})$

Step 4

Rewrite $- 18$ as $- 1 (18)$ .

$(11 - (i \cdot \sqrt{72})) (2 + \sqrt{- 1 (18)})$

Step 5

Rewrite $\sqrt{- 1 (18)}$ as $\sqrt{- 1} \cdot \sqrt{18}$ .

$(11 - (i \cdot \sqrt{72})) (2 + \sqrt{- 1} \cdot \sqrt{18})$

Step 6

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

$(11 - (i \cdot \sqrt{72})) (2 + i \cdot \sqrt{18})$

Step 7

Simplify terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

Rewrite $72$ as $6^{2} \cdot 2$ .

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Step 7.1.1.1

Factor $36$ out of $72$ .

$(11 - (i \cdot \sqrt{36 (2)})) (2 + i \cdot \sqrt{18})$

Step 7.1.1.2

Rewrite $36$ as $6^{2}$ .

$(11 - (i \cdot \sqrt{6^{2} \cdot 2})) (2 + i \cdot \sqrt{18})$

Step 7.1.2

Pull terms out from under the radical.

$(11 - (i \cdot (6 \sqrt{2}))) (2 + i \cdot \sqrt{18})$

Step 7.1.3

Move $6$ to the left of $i$ .

$(11 - (6 \cdot i \sqrt{2})) (2 + i \cdot \sqrt{18})$

Step 7.1.4

Multiply $6$ by $- 1$ .

$(11 - 6 i \sqrt{2}) (2 + i \cdot \sqrt{18})$

Step 7.2

Simplify each term.

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Step 7.2.1

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 7.2.1.1

Factor $9$ out of $18$ .

$(11 - 6 i \sqrt{2}) (2 + i \cdot \sqrt{9 (2)})$

Step 7.2.1.2

Rewrite $9$ as $3^{2}$ .

$(11 - 6 i \sqrt{2}) (2 + i \cdot \sqrt{3^{2} \cdot 2})$

Step 7.2.2

Pull terms out from under the radical.

$(11 - 6 i \sqrt{2}) (2 + i \cdot (3 \sqrt{2}))$

Step 7.2.3

Move $3$ to the left of $i$ .

$(11 - 6 i \sqrt{2}) (2 + 3 i \sqrt{2})$

Step 8

Expand $(11 - 6 i \sqrt{2}) (2 + 3 i \sqrt{2})$ using the FOIL Method.

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Step 8.1

Apply the distributive property.

$11 (2 + 3 i \sqrt{2}) - 6 i \sqrt{2} (2 + 3 i \sqrt{2})$

Step 8.2

Apply the distributive property.

$11 \cdot 2 + 11 (3 i \sqrt{2}) - 6 i \sqrt{2} (2 + 3 i \sqrt{2})$

Step 8.3

Apply the distributive property.

$11 \cdot 2 + 11 (3 i \sqrt{2}) - 6 i \sqrt{2} \cdot 2 - 6 i \sqrt{2} (3 i \sqrt{2})$

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 $11$ by $2$ .

$22 + 11 (3 i \sqrt{2}) - 6 i \sqrt{2} \cdot 2 - 6 i \sqrt{2} (3 i \sqrt{2})$

Step 9.1.2

Multiply $3$ by $11$ .

$22 + 33 (i \sqrt{2}) - 6 i \sqrt{2} \cdot 2 - 6 i \sqrt{2} (3 i \sqrt{2})$

Step 9.1.3

Multiply $2$ by $- 6$ .

$22 + 33 i \sqrt{2} - 12 i \sqrt{2} - 6 i \sqrt{2} (3 i \sqrt{2})$

Step 9.1.4

Multiply $- 6 i \sqrt{2} (3 i \sqrt{2})$ .

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Step 9.1.4.1

Multiply $3$ by $- 6$ .

$22 + 33 i \sqrt{2} - 12 i \sqrt{2} - 18 i \sqrt{2} (i \sqrt{2})$

Step 9.1.4.2

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

$22 + 33 i \sqrt{2} - 12 i \sqrt{2} - 18 (i^{1} i) \sqrt{2} \sqrt{2}$

Step 9.1.4.3

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

$22 + 33 i \sqrt{2} - 12 i \sqrt{2} - 18 (i^{1} i^{1}) \sqrt{2} \sqrt{2}$

Step 9.1.4.4

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

$22 + 33 i \sqrt{2} - 12 i \sqrt{2} - 18 i^{1 + 1} \sqrt{2} \sqrt{2}$

Step 9.1.4.5

Add $1$ and $1$ .

$22 + 33 i \sqrt{2} - 12 i \sqrt{2} - 18 i^{2} \sqrt{2} \sqrt{2}$

Step 9.1.4.6

Raise $\sqrt{2}$ to the power of $1$ .

$22 + 33 i \sqrt{2} - 12 i \sqrt{2} - 18 i^{2} ({\sqrt{2}}^{1} \sqrt{2})$

Step 9.1.4.7

Raise $\sqrt{2}$ to the power of $1$ .

$22 + 33 i \sqrt{2} - 12 i \sqrt{2} - 18 i^{2} ({\sqrt{2}}^{1} {\sqrt{2}}^{1})$

Step 9.1.4.8

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

$22 + 33 i \sqrt{2} - 12 i \sqrt{2} - 18 i^{2} {\sqrt{2}}^{1 + 1}$

Step 9.1.4.9

Add $1$ and $1$ .

$22 + 33 i \sqrt{2} - 12 i \sqrt{2} - 18 i^{2} {\sqrt{2}}^{2}$

Step 9.1.5

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

$22 + 33 i \sqrt{2} - 12 i \sqrt{2} - 18 \cdot - 1 {\sqrt{2}}^{2}$

Step 9.1.6

Multiply $- 18$ by $- 1$ .

$22 + 33 i \sqrt{2} - 12 i \sqrt{2} + 18 {\sqrt{2}}^{2}$

Step 9.1.7

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

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Step 9.1.7.1

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

$22 + 33 i \sqrt{2} - 12 i \sqrt{2} + 18 {(2^{\frac{1}{2}})}^{2}$

Step 9.1.7.2

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

$22 + 33 i \sqrt{2} - 12 i \sqrt{2} + 18 \cdot 2^{\frac{1}{2} \cdot 2}$

Step 9.1.7.3

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

$22 + 33 i \sqrt{2} - 12 i \sqrt{2} + 18 \cdot 2^{\frac{2}{2}}$

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.

$22 + 33 i \sqrt{2} - 12 i \sqrt{2} + 18 \cdot 2^{\frac{2}{2}}$

Step 9.1.7.4.2

Rewrite the expression.

$22 + 33 i \sqrt{2} - 12 i \sqrt{2} + 18 \cdot 2^{1}$

Step 9.1.7.5

Evaluate the exponent.

$22 + 33 i \sqrt{2} - 12 i \sqrt{2} + 18 \cdot 2$

Step 9.1.8

Multiply $18$ by $2$ .

$22 + 33 i \sqrt{2} - 12 i \sqrt{2} + 36$

Step 9.2

Add $22$ and $36$ .

$33 i \sqrt{2} - 12 i \sqrt{2} + 58$

Step 9.3

Subtract $12 i \sqrt{2}$ from $33 i \sqrt{2}$ .

$21 i \sqrt{2} + 58$

Step 10

Reorder $21 i \sqrt{2}$ and $58$ .

$58 + 21 i \sqrt{2}$