Precalculus Examples

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

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

Step 1

Rewrite

- 7

$- 7$ as

- 1 (7)

$- 1 (7)$ .

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

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

Step 2

Rewrite

\sqrt{- 1 (7)}

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

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

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

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

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

Step 3

Rewrite

\sqrt{- 1}

$\sqrt{- 1}$ as

i

$i$ .

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

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

Step 4

Rewrite

- 28

$- 28$ as

- 1 (28)

$- 1 (28)$ .

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

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

Step 5

Rewrite

\sqrt{- 1 (28)}

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

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

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

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

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

Step 6

Rewrite

\sqrt{- 1}

$\sqrt{- 1}$ as

i

$i$ .

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

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

Step 7

Rewrite

28

$28$ as

2^{2} \cdot 7

$2^{2} \cdot 7$ .

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Step 7.1

Factor

4

$4$ out of

28

$28$ .

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

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

Step 7.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

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

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

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

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

Step 8

Pull terms out from under the radical.

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

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

Step 9

Move

2

$2$ to the left of

i

$i$ .

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

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

Step 10

Multiply

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

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

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Step 10.1

Raise

i

$i$ to the power of

1

$1$ .

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

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

Step 10.2

Raise

i

$i$ to the power of

1

$1$ .

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

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

Step 10.3

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 10.4

Add

1

$1$ and

1

$1$ .

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

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

Step 10.5

Raise

\sqrt{7}

$\sqrt{7}$ to the power of

1

$1$ .

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

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

Step 10.6

Raise

\sqrt{7}

$\sqrt{7}$ to the power of

1

$1$ .

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

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

Step 10.7

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 10.8

Add

1

$1$ and

1

$1$ .

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

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

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

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

Step 11

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

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

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

Step 12

Multiply

2

$2$ by

- 1

$- 1$ .

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

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

Step 13

Rewrite

{\sqrt{7}}^{2}

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

7

$7$ .

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Step 13.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{7}

$\sqrt{7}$ as

7^{\frac{1}{2}}

$7^{\frac{1}{2}}$ .

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

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

Step 13.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 13.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 13.4

Cancel the common factor of

2

$2$ .

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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.

$- 2 \cdot 7^{1}$

$- 2 \cdot 7^{1}$

Step 13.5

Evaluate the exponent.

$- 2 \cdot 7$

$- 2 \cdot 7$

Step 14

Multiply

$- 2$ by

$7$ .

$- 14$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$