Solve by Factoring square root of x+16=3 square root of x

$\sqrt{x + 16} = 3 \sqrt{x}$

Step 1

Subtract $3 \sqrt{x}$ from both sides of the equation.

$\sqrt{x + 16} - 3 \sqrt{x} = 0$

Step 2

Add $3 \sqrt{x}$ to both sides of the equation.

$\sqrt{x + 16} = 3 \sqrt{x}$

Step 3

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

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

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 16}$ as ${(x + 16)}^{\frac{1}{2}}$ .

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

Simplify the left side.

Simplify ${({(x + 16)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x + 16)}^{\frac{1}{2}})}^{2}$ .

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

${(x + 16)}^{\frac{1}{2} \cdot 2} = {(3 \sqrt{x})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x + 16)}^{\frac{1}{2} \cdot 2} = {(3 \sqrt{x})}^{2}$

Rewrite the expression.

${(x + 16)}^{1} = {(3 \sqrt{x})}^{2}$

Simplify.

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

Simplify the right side.

Simplify ${(3 \sqrt{x})}^{2}$ .

Simplify the expression.

Apply the product rule to $3 \sqrt{x}$ .

$x + 16 = 3^{2} {\sqrt{x}}^{2}$

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

$x + 16 = 9 {\sqrt{x}}^{2}$

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

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

$x + 16 = 9 {(x^{\frac{1}{2}})}^{2}$

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

$x + 16 = 9 x^{\frac{1}{2} \cdot 2}$

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

$x + 16 = 9 x^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x + 16 = 9 x^{\frac{2}{2}}$

Rewrite the expression.

$x + 16 = 9 x^{1}$

Simplify.

$x + 16 = 9 x$

Step 5

Solve for $x$ .

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

Subtract $9 x$ from both sides of the equation.

$x + 16 - 9 x = 0$

Subtract $9 x$ from $x$ .

$- 8 x + 16 = 0$

Subtract $16$ from both sides of the equation.

$- 8 x = - 16$

Divide each term in $- 8 x = - 16$ by $- 8$ and simplify.

Divide each term in $- 8 x = - 16$ by $- 8$ .

$\frac{- 8 x}{- 8} = \frac{- 16}{- 8}$

Simplify the left side.

Cancel the common factor of $- 8$ .

Cancel the common factor.

$\frac{- 8 x}{- 8} = \frac{- 16}{- 8}$

Divide $x$ by $1$ .

$x = \frac{- 16}{- 8}$

Simplify the right side.

Divide $- 16$ by $- 8$ .

$x = 2$

Solve by Factoring square root of x+16=3 square root of x

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