Finite Math Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

Factor $9$ out of $27$ .

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

Step 4.2

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

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

Step 5

Pull terms out from under the radical.

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

Step 6

Move $3$ to the left of $i$ .

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

Step 7

Multiply $\sqrt{3} (3 i \sqrt{3})$ .

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

Add $1$ and $1$ .

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

Cancel the common factor of $2$ .

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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.

$3 i \cdot 3^{1}$

Step 8.5

Evaluate the exponent.

$3 i \cdot 3$

Step 9

Multiply $3$ by $3$ .

$9 i$