Algebra Examples

\sqrt{- 10} \cdot \sqrt{- 24}

$\sqrt{- 10} \cdot \sqrt{- 24}$

Step 1

Rewrite

- 10

$- 10$ as

- 1 (10)

$- 1 (10)$ .

\sqrt{- 1 (10)} \cdot \sqrt{- 24}

$\sqrt{- 1 (10)} \cdot \sqrt{- 24}$

Step 2

Rewrite

\sqrt{- 1 (10)}

$\sqrt{- 1 (10)}$ as

\sqrt{- 1} \cdot \sqrt{10}

$\sqrt{- 1} \cdot \sqrt{10}$ .

\sqrt{- 1} \cdot \sqrt{10} \cdot \sqrt{- 24}

$\sqrt{- 1} \cdot \sqrt{10} \cdot \sqrt{- 24}$

Step 3

Rewrite

\sqrt{- 1}

$\sqrt{- 1}$ as

i

$i$ .

i \cdot \sqrt{10} \cdot \sqrt{- 24}

$i \cdot \sqrt{10} \cdot \sqrt{- 24}$

Step 4

Rewrite

- 24

$- 24$ as

- 1 (24)

$- 1 (24)$ .

i \cdot \sqrt{10} \cdot \sqrt{- 1 (24)}

$i \cdot \sqrt{10} \cdot \sqrt{- 1 (24)}$

Step 5

Rewrite

\sqrt{- 1 (24)}

$\sqrt{- 1 (24)}$ as

\sqrt{- 1} \cdot \sqrt{24}

$\sqrt{- 1} \cdot \sqrt{24}$ .

i \cdot \sqrt{10} \cdot (\sqrt{- 1} \cdot \sqrt{24})

$i \cdot \sqrt{10} \cdot (\sqrt{- 1} \cdot \sqrt{24})$

Step 6

Rewrite

\sqrt{- 1}

$\sqrt{- 1}$ as

i

$i$ .

i \cdot \sqrt{10} \cdot (i \cdot \sqrt{24})

$i \cdot \sqrt{10} \cdot (i \cdot \sqrt{24})$

Step 7

Rewrite

24

$24$ as

2^{2} \cdot 6

$2^{2} \cdot 6$ .

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Step 7.1

Factor

4

$4$ out of

24

$24$ .

i \sqrt{10} \cdot (i \cdot \sqrt{4 (6)})

$i \sqrt{10} \cdot (i \cdot \sqrt{4 (6)})$

Step 7.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

i \sqrt{10} \cdot (i \cdot \sqrt{2^{2} \cdot 6})

$i \sqrt{10} \cdot (i \cdot \sqrt{2^{2} \cdot 6})$

i \sqrt{10} \cdot (i \cdot \sqrt{2^{2} \cdot 6})

$i \sqrt{10} \cdot (i \cdot \sqrt{2^{2} \cdot 6})$

Step 8

Pull terms out from under the radical.

i \sqrt{10} \cdot (i \cdot (2 \sqrt{6}))

$i \sqrt{10} \cdot (i \cdot (2 \sqrt{6}))$

Step 9

Move

2

$2$ to the left of

i

$i$ .

i \sqrt{10} \cdot (2 \cdot i \sqrt{6})

$i \sqrt{10} \cdot (2 \cdot i \sqrt{6})$

Step 10

Multiply

i \sqrt{10} (2 i \sqrt{6})

$i \sqrt{10} (2 i \sqrt{6})$ .

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Step 10.1

Raise

i

$i$ to the power of

1

$1$ .

\sqrt{10} (2 (i^{1} i) \sqrt{6})

$\sqrt{10} (2 (i^{1} i) \sqrt{6})$

Step 10.2

Raise

i

$i$ to the power of

1

$1$ .

\sqrt{10} (2 (i^{1} i^{1}) \sqrt{6})

$\sqrt{10} (2 (i^{1} i^{1}) \sqrt{6})$

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{10} (2 i^{1 + 1} \sqrt{6})

$\sqrt{10} (2 i^{1 + 1} \sqrt{6})$

Step 10.4

Add

1

$1$ and

1

$1$ .

\sqrt{10} (2 i^{2} \sqrt{6})

$\sqrt{10} (2 i^{2} \sqrt{6})$

Step 10.5

Combine using the product rule for radicals.

2 i^{2} \sqrt{10 \cdot 6}

$2 i^{2} \sqrt{10 \cdot 6}$

Step 10.6

Multiply

10

$10$ by

6

$6$ .

2 i^{2} \sqrt{60}

$2 i^{2} \sqrt{60}$

2 i^{2} \sqrt{60}

$2 i^{2} \sqrt{60}$

Step 11

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

2 \cdot - 1 \sqrt{60}

$2 \cdot - 1 \sqrt{60}$

Step 12

Multiply

2

$2$ by

- 1

$- 1$ .

- 2 \sqrt{60}

$- 2 \sqrt{60}$

Step 13

Rewrite

60

$60$ as

2^{2} \cdot 15

$2^{2} \cdot 15$ .

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Step 13.1

Factor

4

$4$ out of

60

$60$ .

- 2 \sqrt{4 (15)}

$- 2 \sqrt{4 (15)}$

Step 13.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

- 2 \sqrt{2^{2} \cdot 15}

$- 2 \sqrt{2^{2} \cdot 15}$

- 2 \sqrt{2^{2} \cdot 15}

$- 2 \sqrt{2^{2} \cdot 15}$

Step 14

Pull terms out from under the radical.

- 2 (2 \sqrt{15})

$- 2 (2 \sqrt{15})$

Step 15

Multiply

2

$2$ by

- 2

$- 2$ .

- 4 \sqrt{15}

$- 4 \sqrt{15}$

Step 16

The result can be shown in multiple forms.

Exact Form:

- 4 \sqrt{15}

$- 4 \sqrt{15}$

Decimal Form:

- 15.49193338 \dots

$- 15.49193338 \dots$

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

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