Solve for h d = square root of (3h)/2

$d = \sqrt{\frac{3 h}{2}}$

Step 1

Rewrite the equation as $\sqrt{\frac{3 h}{2}} = d$ .

$\sqrt{\frac{3 h}{2}} = d$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{\frac{3 h}{2}}}^{2} = d^{2}$

Step 3

Simplify each side of the equation.

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

${({(\frac{3 h}{2})}^{\frac{1}{2}})}^{2} = d^{2}$

Simplify the left side.

Simplify ${({(\frac{3 h}{2})}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(\frac{3 h}{2})}^{\frac{1}{2}})}^{2}$ .

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

${(\frac{3 h}{2})}^{\frac{1}{2} \cdot 2} = d^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(\frac{3 h}{2})}^{\frac{1}{2} \cdot 2} = d^{2}$

Rewrite the expression.

${(\frac{3 h}{2})}^{1} = d^{2}$

Simplify.

$\frac{3 h}{2} = d^{2}$

Step 4

Solve for $h$ .

Multiply both sides of the equation by $\frac{2}{3}$ .

$\frac{2}{3} \cdot \frac{3 h}{2} = \frac{2}{3} d^{2}$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{2}{3} \cdot \frac{3 h}{2}$ .

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2}{3} \cdot \frac{3 h}{2} = \frac{2}{3} d^{2}$

Rewrite the expression.

$\frac{1}{3} (3 h) = \frac{2}{3} d^{2}$

Cancel the common factor of $3$ .

Factor $3$ out of $3 h$ .

$\frac{1}{3} (3 (h)) = \frac{2}{3} d^{2}$

Cancel the common factor.

$\frac{1}{3} (3 h) = \frac{2}{3} d^{2}$

Rewrite the expression.

$h = \frac{2}{3} d^{2}$

Simplify the right side.

Combine $\frac{2}{3}$ and $d^{2}$ .

$h = \frac{2 d^{2}}{3}$