Precalculus Examples

$\sqrt{- 7} \sqrt{- 28}$

Step 1

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

$\sqrt{- 1 (7)} \sqrt{- 28}$

Step 2

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

$\sqrt{- 1} \cdot \sqrt{7} \sqrt{- 28}$

Step 3

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

$i \cdot \sqrt{7} \sqrt{- 28}$

Step 4

Rewrite $- 28$ as $- 1 (28)$ .

$i \cdot \sqrt{7} \sqrt{- 1 (28)}$

Step 5

Rewrite $\sqrt{- 1 (28)}$ as $\sqrt{- 1} \cdot \sqrt{28}$ .

$i \cdot \sqrt{7} (\sqrt{- 1} \cdot \sqrt{28})$

Step 6

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

$i \cdot \sqrt{7} (i \cdot \sqrt{28})$

Step 7

Rewrite $28$ as $2^{2} \cdot 7$ .

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Step 7.1

Factor $4$ out of $28$ .

$i \sqrt{7} (i \cdot \sqrt{4 (7)})$

Step 7.2

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

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

Step 8

Pull terms out from under the radical.

$i \sqrt{7} (i \cdot (2 \sqrt{7}))$

Step 9

Move $2$ to the left of $i$ .

$i \sqrt{7} (2 \cdot i \sqrt{7})$

Step 10

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

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

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

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

Step 10.4

Add $1$ and $1$ .

$\sqrt{7} (2 i^{2} \sqrt{7})$

Step 10.5

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

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

Step 10.6

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

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

Step 10.7

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

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

Step 10.8

Add $1$ and $1$ .

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

Step 11

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

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

Step 12

Multiply $2$ by $- 1$ .

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

Step 13

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

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Step 13.1

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

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

Step 13.2

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

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

Step 13.3

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

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

Step 13.4

Cancel the common factor of $2$ .

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Step 13.4.1

Cancel the common factor.

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

Step 13.4.2

Rewrite the expression.

$- 2 \cdot 7^{1}$

Step 13.5

Evaluate the exponent.

$- 2 \cdot 7$

Step 14

Multiply $- 2$ by $7$ .

$- 14$