Precalculus Examples

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

Step 1

Rewrite $- 27$ as $- 1 (27)$ .

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

Step 2

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

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

Step 3

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

$\sqrt{\frac{1}{3}} (i \cdot \sqrt{27})$

Step 4

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$\frac{\sqrt{1}}{\sqrt{3}} (i \cdot \sqrt{27})$

Step 5

Any root of $1$ is $1$ .

$\frac{1}{\sqrt{3}} (i \cdot \sqrt{27})$

Step 6

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} (i \cdot \sqrt{27})$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\sqrt{3}}{\sqrt{3} \sqrt{3}} (i \cdot \sqrt{27})$

Step 7.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} (i \cdot \sqrt{27})$

Step 7.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} (i \cdot \sqrt{27})$

Step 7.4

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

$\frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}} (i \cdot \sqrt{27})$

Step 7.5

Add $1$ and $1$ .

$\frac{\sqrt{3}}{{\sqrt{3}}^{2}} (i \cdot \sqrt{27})$

Step 7.6

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

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Step 7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} (i \cdot \sqrt{27})$

Step 7.6.2

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

$\frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}} (i \cdot \sqrt{27})$

Step 7.6.3

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

$\frac{\sqrt{3}}{3^{\frac{2}{2}}} (i \cdot \sqrt{27})$

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

$\frac{\sqrt{3}}{3^{\frac{2}{2}}} (i \cdot \sqrt{27})$

Step 7.6.4.2

Rewrite the expression.

$\frac{\sqrt{3}}{3^{1}} (i \cdot \sqrt{27})$

Step 7.6.5

Evaluate the exponent.

$\frac{\sqrt{3}}{3} (i \cdot \sqrt{27})$

Step 8

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 8.1

Factor $9$ out of $27$ .

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

Step 8.2

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

$\frac{\sqrt{3}}{3} (i \cdot \sqrt{3^{2} \cdot 3})$

Step 9

Pull terms out from under the radical.

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

Step 10

Cancel the common factor of $3$ .

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Step 10.1

Factor $3$ out of $i \cdot (3 \sqrt{3})$ .

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

Step 10.2

Cancel the common factor.

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

Step 10.3

Rewrite the expression.

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

Step 11

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 12

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 13

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

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

Step 14

Add $1$ and $1$ .

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

Step 15

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

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Step 15.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 15.2

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

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

Step 15.3

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

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

Step 15.4

Cancel the common factor of $2$ .

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Step 15.4.1

Cancel the common factor.

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

Step 15.4.2

Rewrite the expression.

$3^{1} i$

Step 15.5

Evaluate the exponent.

$3 i$