Simplify square root of -4/3

$\sqrt{- \frac{4}{3}}$

Step 1

Rewrite $- \frac{4}{3}$ as $i^{2} \frac{2^{2}}{3}$ .

Rewrite $- 1$ as $i^{2}$ .

$\sqrt{i^{2} \frac{4}{3}}$

Rewrite $4$ as $2^{2}$ .

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

Step 2

Pull terms out from under the radical.

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

Step 3

Raise $2$ to the power of $2$ .

$i \sqrt{\frac{4}{3}}$

Step 4

Rewrite $\sqrt{\frac{4}{3}}$ as $\frac{\sqrt{4}}{\sqrt{3}}$ .

$i \frac{\sqrt{4}}{\sqrt{3}}$

Step 5

Simplify the numerator.

Rewrite $4$ as $2^{2}$ .

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

Pull terms out from under the radical, assuming positive real numbers.

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

Step 6

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 7

Combine and simplify the denominator.

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 8

Combine $i$ and $\frac{2 \sqrt{3}}{3}$ .

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

Step 9

Move $2$ to the left of $i$ .

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

Simplify square root of -4/3

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