Solve for r 3 square root of r = square root of 8r+16

$3 \sqrt{r} = \sqrt{8 r + 16}$

Step 1

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

${(3 \sqrt{r})}^{2} = {\sqrt{8 r + 16}}^{2}$

Step 2

Simplify each side of the equation.

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

${(3 r^{\frac{1}{2}})}^{2} = {\sqrt{8 r + 16}}^{2}$

Simplify the left side.

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

Apply the product rule to $3 r^{\frac{1}{2}}$ .

$3^{2} {(r^{\frac{1}{2}})}^{2} = {\sqrt{8 r + 16}}^{2}$

Raise $3$ to the power of $2$ .

$9 {(r^{\frac{1}{2}})}^{2} = {\sqrt{8 r + 16}}^{2}$

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

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

$9 r^{\frac{1}{2} \cdot 2} = {\sqrt{8 r + 16}}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$9 r^{\frac{1}{2} \cdot 2} = {\sqrt{8 r + 16}}^{2}$

Rewrite the expression.

$9 r^{1} = {\sqrt{8 r + 16}}^{2}$

Simplify.

$9 r = {\sqrt{8 r + 16}}^{2}$

Simplify the right side.

Simplify ${\sqrt{8 r + 16}}^{2}$ .

Factor $8$ out of $8 r + 16$ .

Factor $8$ out of $8 r$ .

$9 r = {\sqrt{8 (r) + 16}}^{2}$

Factor $8$ out of $16$ .

$9 r = {\sqrt{8 r + 8 \cdot 2}}^{2}$

Factor $8$ out of $8 r + 8 \cdot 2$ .

$9 r = {\sqrt{8 (r + 2)}}^{2}$

Rewrite $8 (r + 2)$ as $2^{2} \cdot (2 (r + 2))$ .

Factor $4$ out of $8$ .

$9 r = {\sqrt{4 (2) (r + 2)}}^{2}$

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

$9 r = {\sqrt{2^{2} \cdot 2 (r + 2)}}^{2}$

Add parentheses.

$9 r = {\sqrt{2^{2} \cdot (2 (r + 2))}}^{2}$

Pull terms out from under the radical.

$9 r = {(2 \sqrt{2 (r + 2)})}^{2}$

Simplify the expression.

Apply the product rule to $2 \sqrt{2 (r + 2)}$ .

$9 r = 2^{2} {\sqrt{2 (r + 2)}}^{2}$

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

$9 r = 4 {\sqrt{2 (r + 2)}}^{2}$

Rewrite ${\sqrt{2 (r + 2)}}^{2}$ as $2 (r + 2)$ .

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

$9 r = 4 {({(2 (r + 2))}^{\frac{1}{2}})}^{2}$

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

$9 r = 4 {(2 (r + 2))}^{\frac{1}{2} \cdot 2}$

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

$9 r = 4 {(2 (r + 2))}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$9 r = 4 {(2 (r + 2))}^{\frac{2}{2}}$

Rewrite the expression.

$9 r = 4 {(2 (r + 2))}^{1}$

Simplify.

$9 r = 4 (2 (r + 2))$

Apply the distributive property.

$9 r = 4 (2 r + 2 \cdot 2)$

Multiply $2$ by $2$ .

$9 r = 4 (2 r + 4)$

Apply the distributive property.

$9 r = 4 (2 r) + 4 \cdot 4$

Multiply.

Multiply $2$ by $4$ .

$9 r = 8 r + 4 \cdot 4$

Multiply $4$ by $4$ .

$9 r = 8 r + 16$

Step 3

Move all terms containing $r$ to the left side of the equation.

Subtract $8 r$ from both sides of the equation.

$9 r - 8 r = 16$

Subtract $8 r$ from $9 r$ .

$r = 16$

Solve for r 3 square root of r = square root of 8r+16

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