Simplify ( square root of 25)/( square root of -75)

$\frac{\sqrt{25}}{\sqrt{- 75}}$

Step 1

Pull out imaginary unit $i$ .

Rewrite $- 75$ as $- 1 (75)$ .

$\frac{\sqrt{25}}{\sqrt{- 1 (75)}}$

Rewrite $\sqrt{- 1 (75)}$ as $\sqrt{- 1} \cdot \sqrt{75}$ .

$\frac{\sqrt{25}}{\sqrt{- 1} \cdot \sqrt{75}}$

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

$\frac{\sqrt{25}}{i \cdot \sqrt{75}}$

Step 2

Combine $\sqrt{25}$ and $\sqrt{75}$ into a single radical.

$\frac{\sqrt{\frac{25}{75}}}{i}$

Step 3

Cancel the common factor of $25$ and $75$ .

Factor $25$ out of $25$ .

$\frac{\sqrt{\frac{25 (1)}{75}}}{i}$

Cancel the common factors.

Factor $25$ out of $75$ .

$\frac{\sqrt{\frac{25 \cdot 1}{25 \cdot 3}}}{i}$

Cancel the common factor.

$\frac{\sqrt{\frac{25 \cdot 1}{25 \cdot 3}}}{i}$

Rewrite the expression.

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

Step 4

Simplify the numerator.

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

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

Any root of $1$ is $1$ .

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

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

$\frac{\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}}{i}$

Combine and simplify the denominator.

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

$\frac{\frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}}{i}$

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

$\frac{\frac{\sqrt{3}}{3}}{i}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{3}}{3} \cdot \frac{1}{i}$

Step 6

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

$\frac{\sqrt{3}}{3 i}$

Simplify ( square root of 25)/( square root of -75)

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