Algebra Examples

$(6 + \sqrt{- 98}) (11 + \sqrt{- 32})$

Step 1

Rewrite $- 98$ as $- 1 (98)$ .

$(6 + \sqrt{- 1 (98)}) (11 + \sqrt{- 32})$

Step 2

Rewrite $\sqrt{- 1 (98)}$ as $\sqrt{- 1} \cdot \sqrt{98}$ .

$(6 + \sqrt{- 1} \cdot \sqrt{98}) (11 + \sqrt{- 32})$

Step 3

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

$(6 + i \cdot \sqrt{98}) (11 + \sqrt{- 32})$

Step 4

Rewrite $- 32$ as $- 1 (32)$ .

$(6 + i \cdot \sqrt{98}) (11 + \sqrt{- 1 (32)})$

Step 5

Rewrite $\sqrt{- 1 (32)}$ as $\sqrt{- 1} \cdot \sqrt{32}$ .

$(6 + i \cdot \sqrt{98}) (11 + \sqrt{- 1} \cdot \sqrt{32})$

Step 6

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

$(6 + i \cdot \sqrt{98}) (11 + i \cdot \sqrt{32})$

Step 7

Simplify terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

Rewrite $98$ as $7^{2} \cdot 2$ .

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Step 7.1.1.1

Factor $49$ out of $98$ .

$(6 + i \cdot \sqrt{49 (2)}) (11 + i \cdot \sqrt{32})$

Step 7.1.1.2

Rewrite $49$ as $7^{2}$ .

$(6 + i \cdot \sqrt{7^{2} \cdot 2}) (11 + i \cdot \sqrt{32})$

Step 7.1.2

Pull terms out from under the radical.

$(6 + i \cdot (7 \sqrt{2})) (11 + i \cdot \sqrt{32})$

Step 7.1.3

Move $7$ to the left of $i$ .

$(6 + 7 i \sqrt{2}) (11 + i \cdot \sqrt{32})$

Step 7.2

Simplify each term.

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Step 7.2.1

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 7.2.1.1

Factor $16$ out of $32$ .

$(6 + 7 i \sqrt{2}) (11 + i \cdot \sqrt{16 (2)})$

Step 7.2.1.2

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

$(6 + 7 i \sqrt{2}) (11 + i \cdot \sqrt{4^{2} \cdot 2})$

Step 7.2.2

Pull terms out from under the radical.

$(6 + 7 i \sqrt{2}) (11 + i \cdot (4 \sqrt{2}))$

Step 7.2.3

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

$(6 + 7 i \sqrt{2}) (11 + 4 i \sqrt{2})$

Step 8

Expand $(6 + 7 i \sqrt{2}) (11 + 4 i \sqrt{2})$ using the FOIL Method.

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Step 8.1

Apply the distributive property.

$6 (11 + 4 i \sqrt{2}) + 7 i \sqrt{2} (11 + 4 i \sqrt{2})$

Step 8.2

Apply the distributive property.

$6 \cdot 11 + 6 (4 i \sqrt{2}) + 7 i \sqrt{2} (11 + 4 i \sqrt{2})$

Step 8.3

Apply the distributive property.

$6 \cdot 11 + 6 (4 i \sqrt{2}) + 7 i \sqrt{2} \cdot 11 + 7 i \sqrt{2} (4 i \sqrt{2})$

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 $6$ by $11$ .

$66 + 6 (4 i \sqrt{2}) + 7 i \sqrt{2} \cdot 11 + 7 i \sqrt{2} (4 i \sqrt{2})$

Step 9.1.2

Multiply $4$ by $6$ .

$66 + 24 (i \sqrt{2}) + 7 i \sqrt{2} \cdot 11 + 7 i \sqrt{2} (4 i \sqrt{2})$

Step 9.1.3

Multiply $11$ by $7$ .

$66 + 24 i \sqrt{2} + 77 i \sqrt{2} + 7 i \sqrt{2} (4 i \sqrt{2})$

Step 9.1.4

Multiply $7 i \sqrt{2} (4 i \sqrt{2})$ .

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Step 9.1.4.1

Multiply $4$ by $7$ .

$66 + 24 i \sqrt{2} + 77 i \sqrt{2} + 28 i \sqrt{2} (i \sqrt{2})$

Step 9.1.4.2

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

$66 + 24 i \sqrt{2} + 77 i \sqrt{2} + 28 (i^{1} i) \sqrt{2} \sqrt{2}$

Step 9.1.4.3

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

$66 + 24 i \sqrt{2} + 77 i \sqrt{2} + 28 (i^{1} i^{1}) \sqrt{2} \sqrt{2}$

Step 9.1.4.4

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

$66 + 24 i \sqrt{2} + 77 i \sqrt{2} + 28 i^{1 + 1} \sqrt{2} \sqrt{2}$

Step 9.1.4.5

Add $1$ and $1$ .

$66 + 24 i \sqrt{2} + 77 i \sqrt{2} + 28 i^{2} \sqrt{2} \sqrt{2}$

Step 9.1.4.6

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

$66 + 24 i \sqrt{2} + 77 i \sqrt{2} + 28 i^{2} ({\sqrt{2}}^{1} \sqrt{2})$

Step 9.1.4.7

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

$66 + 24 i \sqrt{2} + 77 i \sqrt{2} + 28 i^{2} ({\sqrt{2}}^{1} {\sqrt{2}}^{1})$

Step 9.1.4.8

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

$66 + 24 i \sqrt{2} + 77 i \sqrt{2} + 28 i^{2} {\sqrt{2}}^{1 + 1}$

Step 9.1.4.9

Add $1$ and $1$ .

$66 + 24 i \sqrt{2} + 77 i \sqrt{2} + 28 i^{2} {\sqrt{2}}^{2}$

Step 9.1.5

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

$66 + 24 i \sqrt{2} + 77 i \sqrt{2} + 28 \cdot - 1 {\sqrt{2}}^{2}$

Step 9.1.6

Multiply $28$ by $- 1$ .

$66 + 24 i \sqrt{2} + 77 i \sqrt{2} - 28 {\sqrt{2}}^{2}$

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 $2^{\frac{1}{2}}$ .

$66 + 24 i \sqrt{2} + 77 i \sqrt{2} - 28 {(2^{\frac{1}{2}})}^{2}$

Step 9.1.7.2

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

$66 + 24 i \sqrt{2} + 77 i \sqrt{2} - 28 \cdot 2^{\frac{1}{2} \cdot 2}$

Step 9.1.7.3

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

$66 + 24 i \sqrt{2} + 77 i \sqrt{2} - 28 \cdot 2^{\frac{2}{2}}$

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.

$66 + 24 i \sqrt{2} + 77 i \sqrt{2} - 28 \cdot 2^{\frac{2}{2}}$

Step 9.1.7.4.2

Rewrite the expression.

$66 + 24 i \sqrt{2} + 77 i \sqrt{2} - 28 \cdot 2^{1}$

Step 9.1.7.5

Evaluate the exponent.

$66 + 24 i \sqrt{2} + 77 i \sqrt{2} - 28 \cdot 2$

Step 9.1.8

Multiply $- 28$ by $2$ .

$66 + 24 i \sqrt{2} + 77 i \sqrt{2} - 56$

Step 9.2

Subtract $56$ from $66$ .

$24 i \sqrt{2} + 77 i \sqrt{2} + 10$

Step 9.3

Add $24 i \sqrt{2}$ and $77 i \sqrt{2}$ .

$101 i \sqrt{2} + 10$

Step 10

Reorder $101 i \sqrt{2}$ and $10$ .

$10 + 101 i \sqrt{2}$