Basic Math Examples

\sqrt{- 27} \cdot \sqrt{3}

$\sqrt{- 27} \cdot \sqrt{3}$

Step 1

Rewrite

- 27

$- 27$ as

- 1 (27)

$- 1 (27)$ .

\sqrt{- 1 (27)} \cdot \sqrt{3}

$\sqrt{- 1 (27)} \cdot \sqrt{3}$

Step 2

Rewrite

\sqrt{- 1 (27)}

$\sqrt{- 1 (27)}$ as

\sqrt{- 1} \cdot \sqrt{27}

$\sqrt{- 1} \cdot \sqrt{27}$ .

\sqrt{- 1} \cdot \sqrt{27} \cdot \sqrt{3}

$\sqrt{- 1} \cdot \sqrt{27} \cdot \sqrt{3}$

Step 3

Rewrite

\sqrt{- 1}

$\sqrt{- 1}$ as

i

$i$ .

i \cdot \sqrt{27} \cdot \sqrt{3}

$i \cdot \sqrt{27} \cdot \sqrt{3}$

Step 4

Rewrite

27

$27$ as

3^{2} \cdot 3

$3^{2} \cdot 3$ .

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Step 4.1

Factor

9

$9$ out of

27

$27$ .

i \cdot \sqrt{9 (3)} \cdot \sqrt{3}

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

Step 4.2

Rewrite

9

$9$ as

3^{2}

$3^{2}$ .

i \cdot \sqrt{3^{2} \cdot 3} \cdot \sqrt{3}

$i \cdot \sqrt{3^{2} \cdot 3} \cdot \sqrt{3}$

i \cdot \sqrt{3^{2} \cdot 3} \cdot \sqrt{3}

$i \cdot \sqrt{3^{2} \cdot 3} \cdot \sqrt{3}$

Step 5

Pull terms out from under the radical.

i \cdot (3 \sqrt{3}) \cdot \sqrt{3}

$i \cdot (3 \sqrt{3}) \cdot \sqrt{3}$

Step 6

Move

3

$3$ to the left of

i

$i$ .

3 \cdot i \sqrt{3} \cdot \sqrt{3}

$3 \cdot i \sqrt{3} \cdot \sqrt{3}$

Step 7

Multiply

3 i \sqrt{3} \sqrt{3}

$3 i \sqrt{3} \sqrt{3}$ .

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Step 7.1

Raise

\sqrt{3}

$\sqrt{3}$ to the power of

1

$1$ .

3 i ({\sqrt{3}}^{1} \sqrt{3})

$3 i ({\sqrt{3}}^{1} \sqrt{3})$

Step 7.2

Raise

\sqrt{3}

$\sqrt{3}$ to the power of

1

$1$ .

3 i ({\sqrt{3}}^{1} {\sqrt{3}}^{1})

$3 i ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Step 7.3

Use the power rule

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

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

3 i {\sqrt{3}}^{1 + 1}

$3 i {\sqrt{3}}^{1 + 1}$

Step 7.4

Add

1

$1$ and

1

$1$ .

3 i {\sqrt{3}}^{2}

$3 i {\sqrt{3}}^{2}$

3 i {\sqrt{3}}^{2}

$3 i {\sqrt{3}}^{2}$

Step 8

Rewrite

{\sqrt{3}}^{2}

${\sqrt{3}}^{2}$ as

3

$3$ .

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Step 8.1

Use

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

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

\sqrt{3}

$\sqrt{3}$ as

3^{\frac{1}{2}}

$3^{\frac{1}{2}}$ .

3 i {(3^{\frac{1}{2}})}^{2}

$3 i {(3^{\frac{1}{2}})}^{2}$

Step 8.2

Apply the power rule and multiply exponents,

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

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

3 i \cdot 3^{\frac{1}{2} \cdot 2}

$3 i \cdot 3^{\frac{1}{2} \cdot 2}$

Step 8.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

3 i \cdot 3^{\frac{2}{2}}

$3 i \cdot 3^{\frac{2}{2}}$

Step 8.4

Cancel the common factor of

2

$2$ .

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Step 8.4.1

Cancel the common factor.

$3 i \cdot 3^{\frac{2}{2}}$

Step 8.4.2

Rewrite the expression.

$3 i \cdot 3^{1}$

$3 i \cdot 3^{1}$

Step 8.5

Evaluate the exponent.

$3 i \cdot 3$

$3 i \cdot 3$

Step 9

Multiply

$3$ by

$3$ .

$9 i$

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