Algebra Examples

$(- 3 \sqrt{- 63}) (9 \sqrt{- 7})$

Step 1

Rewrite $- 63$ as $- 1 (63)$ .

$- 3 \sqrt{- 1 (63)} (9 \sqrt{- 7})$

Step 2

Rewrite $\sqrt{- 1 (63)}$ as $\sqrt{- 1} \cdot \sqrt{63}$ .

$- 3 (\sqrt{- 1} \cdot \sqrt{63}) (9 \sqrt{- 7})$

Step 3

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

$- 3 (i \cdot \sqrt{63}) (9 \sqrt{- 7})$

Step 4

Rewrite $- 7$ as $- 1 (7)$ .

$- 3 (i \cdot \sqrt{63}) (9 \sqrt{- 1 (7)})$

Step 5

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$- 3 (i \cdot \sqrt{63}) (9 (\sqrt{- 1} \cdot \sqrt{7}))$

Step 6

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

$- 3 (i \cdot \sqrt{63}) (9 (i \cdot \sqrt{7}))$

Step 7

Rewrite $63$ as $3^{2} \cdot 7$ .

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Step 7.1

Factor $9$ out of $63$ .

$- 3 (i \cdot \sqrt{9 (7)}) (9 (i \cdot \sqrt{7}))$

Step 7.2

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

$- 3 (i \cdot \sqrt{3^{2} \cdot 7}) (9 (i \cdot \sqrt{7}))$

Step 8

Pull terms out from under the radical.

$- 3 (i \cdot (| 3 | \sqrt{7})) (9 (i \cdot \sqrt{7}))$

Step 9

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$- 3 (i \cdot (3 \sqrt{7})) (9 (i \cdot \sqrt{7}))$

Step 10

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

$- 3 (3 \cdot i \sqrt{7}) (9 (i \cdot \sqrt{7}))$

Step 11

Multiply $3$ by $- 3$ .

$- 9 (i \sqrt{7}) (9 (i \cdot \sqrt{7}))$

Step 12

Multiply $- 9 i \sqrt{7} (9 i \sqrt{7})$ .

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Step 12.1

Multiply $9$ by $- 9$ .

$- 81 i \sqrt{7} (i \sqrt{7})$

Step 12.2

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

$- 81 (i^{1} i) \sqrt{7} \sqrt{7}$

Step 12.3

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

$- 81 (i^{1} i^{1}) \sqrt{7} \sqrt{7}$

Step 12.4

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

$- 81 i^{1 + 1} \sqrt{7} \sqrt{7}$

Step 12.5

Add $1$ and $1$ .

$- 81 i^{2} \sqrt{7} \sqrt{7}$

Step 12.6

Raise $\sqrt{7}$ to the power of $1$ .

$- 81 i^{2} ({\sqrt{7}}^{1} \sqrt{7})$

Step 12.7

Raise $\sqrt{7}$ to the power of $1$ .

$- 81 i^{2} ({\sqrt{7}}^{1} {\sqrt{7}}^{1})$

Step 12.8

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

$- 81 i^{2} {\sqrt{7}}^{1 + 1}$

Step 12.9

Add $1$ and $1$ .

$- 81 i^{2} {\sqrt{7}}^{2}$

Step 13

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

$- 81 \cdot - 1 {\sqrt{7}}^{2}$

Step 14

Multiply $- 81$ by $- 1$ .

$81 {\sqrt{7}}^{2}$

Step 15

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 15.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$81 {(7^{\frac{1}{2}})}^{2}$

Step 15.2

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

$81 \cdot 7^{\frac{1}{2} \cdot 2}$

Step 15.3

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

$81 \cdot 7^{\frac{2}{2}}$

Step 15.4

Cancel the common factor of $2$ .

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Step 15.4.1

Cancel the common factor.

$81 \cdot 7^{\frac{2}{2}}$

Step 15.4.2

Rewrite the expression.

$81 \cdot 7^{1}$

Step 15.5

Evaluate the exponent.

$81 \cdot 7$

Step 16

Multiply $81$ by $7$ .

$567$