Solve for x square root of 9x+16=8+ square root of x+16

$\sqrt{9 x + 16} = 8 + \sqrt{x + 16}$

Step 1

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

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

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 x + 16}$ as ${(9 x + 16)}^{\frac{1}{2}}$ .

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

Simplify the left side.

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

Multiply the exponents in ${({(9 x + 16)}^{\frac{1}{2}})}^{2}$ .

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

Simplify ${(8 + \sqrt{x + 16})}^{2}$ .

Rewrite ${(8 + \sqrt{x + 16})}^{2}$ as $(8 + \sqrt{x + 16}) (8 + \sqrt{x + 16})$ .

$9 x + 16 = (8 + \sqrt{x + 16}) (8 + \sqrt{x + 16})$

Expand $(8 + \sqrt{x + 16}) (8 + \sqrt{x + 16})$ using the FOIL Method.

Apply the distributive property.

$9 x + 16 = 8 (8 + \sqrt{x + 16}) + \sqrt{x + 16} (8 + \sqrt{x + 16})$

Apply the distributive property.

$9 x + 16 = 8 \cdot 8 + 8 \sqrt{x + 16} + \sqrt{x + 16} (8 + \sqrt{x + 16})$

Apply the distributive property.

$9 x + 16 = 8 \cdot 8 + 8 \sqrt{x + 16} + \sqrt{x + 16} \cdot 8 + \sqrt{x + 16} \sqrt{x + 16}$

Simplify and combine like terms.

Simplify each term.

Multiply $8$ by $8$ .

$9 x + 16 = 64 + 8 \sqrt{x + 16} + \sqrt{x + 16} \cdot 8 + \sqrt{x + 16} \sqrt{x + 16}$

Move $8$ to the left of $\sqrt{x + 16}$ .

$9 x + 16 = 64 + 8 \sqrt{x + 16} + 8 \cdot \sqrt{x + 16} + \sqrt{x + 16} \sqrt{x + 16}$

Multiply $\sqrt{x + 16} \sqrt{x + 16}$ .

Raise $\sqrt{x + 16}$ to the power of $1$ .

$9 x + 16 = 64 + 8 \sqrt{x + 16} + 8 \sqrt{x + 16} + {\sqrt{x + 16}}^{1} \sqrt{x + 16}$

Raise $\sqrt{x + 16}$ to the power of $1$ .

$9 x + 16 = 64 + 8 \sqrt{x + 16} + 8 \sqrt{x + 16} + {\sqrt{x + 16}}^{1} {\sqrt{x + 16}}^{1}$

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

$9 x + 16 = 64 + 8 \sqrt{x + 16} + 8 \sqrt{x + 16} + {\sqrt{x + 16}}^{1 + 1}$

Add $1$ and $1$ .

$9 x + 16 = 64 + 8 \sqrt{x + 16} + 8 \sqrt{x + 16} + {\sqrt{x + 16}}^{2}$

Rewrite ${\sqrt{x + 16}}^{2}$ as $x + 16$ .

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

$9 x + 16 = 64 + 8 \sqrt{x + 16} + 8 \sqrt{x + 16} + {({(x + 16)}^{\frac{1}{2}})}^{2}$

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

$9 x + 16 = 64 + 8 \sqrt{x + 16} + 8 \sqrt{x + 16} + {(x + 16)}^{\frac{1}{2} \cdot 2}$

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

$9 x + 16 = 64 + 8 \sqrt{x + 16} + 8 \sqrt{x + 16} + {(x + 16)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$9 x + 16 = 64 + 8 \sqrt{x + 16} + 8 \sqrt{x + 16} + {(x + 16)}^{\frac{2}{2}}$

Rewrite the expression.

$9 x + 16 = 64 + 8 \sqrt{x + 16} + 8 \sqrt{x + 16} + {(x + 16)}^{1}$

Simplify.

$9 x + 16 = 64 + 8 \sqrt{x + 16} + 8 \sqrt{x + 16} + x + 16$

Add $64$ and $16$ .

$9 x + 16 = 80 + 8 \sqrt{x + 16} + 8 \sqrt{x + 16} + x$

Add $8 \sqrt{x + 16}$ and $8 \sqrt{x + 16}$ .

$9 x + 16 = 80 + 16 \sqrt{x + 16} + x$

Step 3

Solve for $16 \sqrt{x + 16}$ .

Rewrite the equation as $80 + 16 \sqrt{x + 16} + x = 9 x + 16$ .

$80 + 16 \sqrt{x + 16} + x = 9 x + 16$

Move all terms not containing $16 \sqrt{x + 16}$ to the right side of the equation.

Subtract $80$ from both sides of the equation.

$16 \sqrt{x + 16} + x = 9 x + 16 - 80$

Subtract $x$ from both sides of the equation.

$16 \sqrt{x + 16} = 9 x + 16 - 80 - x$

Subtract $x$ from $9 x$ .

$16 \sqrt{x + 16} = 8 x + 16 - 80$

Subtract $80$ from $16$ .

$16 \sqrt{x + 16} = 8 x - 64$

Step 4

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

${(16 \sqrt{x + 16})}^{2} = {(8 x - 64)}^{2}$

Step 5

Simplify each side of the equation.

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

${(16 {(x + 16)}^{\frac{1}{2}})}^{2} = {(8 x - 64)}^{2}$

Simplify the left side.

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

Apply the product rule to $16 {(x + 16)}^{\frac{1}{2}}$ .

$16^{2} {({(x + 16)}^{\frac{1}{2}})}^{2} = {(8 x - 64)}^{2}$

Raise $16$ to the power of $2$ .

$256 {({(x + 16)}^{\frac{1}{2}})}^{2} = {(8 x - 64)}^{2}$

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

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

$256 {(x + 16)}^{\frac{1}{2} \cdot 2} = {(8 x - 64)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$256 {(x + 16)}^{\frac{1}{2} \cdot 2} = {(8 x - 64)}^{2}$

Rewrite the expression.

$256 {(x + 16)}^{1} = {(8 x - 64)}^{2}$

Simplify.

$256 (x + 16) = {(8 x - 64)}^{2}$

Apply the distributive property.

$256 x + 256 \cdot 16 = {(8 x - 64)}^{2}$

Multiply $256$ by $16$ .

$256 x + 4096 = {(8 x - 64)}^{2}$

Simplify the right side.

Simplify ${(8 x - 64)}^{2}$ .

Rewrite ${(8 x - 64)}^{2}$ as $(8 x - 64) (8 x - 64)$ .

$256 x + 4096 = (8 x - 64) (8 x - 64)$

Expand $(8 x - 64) (8 x - 64)$ using the FOIL Method.

Apply the distributive property.

$256 x + 4096 = 8 x (8 x - 64) - 64 (8 x - 64)$

Apply the distributive property.

$256 x + 4096 = 8 x (8 x) + 8 x \cdot - 64 - 64 (8 x - 64)$

Apply the distributive property.

$256 x + 4096 = 8 x (8 x) + 8 x \cdot - 64 - 64 (8 x) - 64 \cdot - 64$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$256 x + 4096 = 8 \cdot 8 x \cdot x + 8 x \cdot - 64 - 64 (8 x) - 64 \cdot - 64$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$256 x + 4096 = 8 \cdot 8 (x \cdot x) + 8 x \cdot - 64 - 64 (8 x) - 64 \cdot - 64$

Multiply $x$ by $x$ .

$256 x + 4096 = 8 \cdot 8 x^{2} + 8 x \cdot - 64 - 64 (8 x) - 64 \cdot - 64$

Multiply $8$ by $8$ .

$256 x + 4096 = 64 x^{2} + 8 x \cdot - 64 - 64 (8 x) - 64 \cdot - 64$

Multiply $- 64$ by $8$ .

$256 x + 4096 = 64 x^{2} - 512 x - 64 (8 x) - 64 \cdot - 64$

Multiply $8$ by $- 64$ .

$256 x + 4096 = 64 x^{2} - 512 x - 512 x - 64 \cdot - 64$

Multiply $- 64$ by $- 64$ .

$256 x + 4096 = 64 x^{2} - 512 x - 512 x + 4096$

Subtract $512 x$ from $- 512 x$ .

$256 x + 4096 = 64 x^{2} - 1024 x + 4096$

Step 6

Solve for $x$ .

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$64 x^{2} - 1024 x + 4096 = 256 x + 4096$

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

Subtract $256 x$ from both sides of the equation.

$64 x^{2} - 1024 x + 4096 - 256 x = 4096$

Subtract $256 x$ from $- 1024 x$ .

$64 x^{2} - 1280 x + 4096 = 4096$

Subtract $4096$ from both sides of the equation.

$64 x^{2} - 1280 x + 4096 - 4096 = 0$

Combine the opposite terms in $64 x^{2} - 1280 x + 4096 - 4096$ .

Subtract $4096$ from $4096$ .

$64 x^{2} - 1280 x + 0 = 0$

Add $64 x^{2} - 1280 x$ and $0$ .

$64 x^{2} - 1280 x = 0$

Factor $64 x$ out of $64 x^{2} - 1280 x$ .

Factor $64 x$ out of $64 x^{2}$ .

$64 x (x) - 1280 x = 0$

Factor $64 x$ out of $- 1280 x$ .

$64 x (x) + 64 x (- 20) = 0$

Factor $64 x$ out of $64 x (x) + 64 x (- 20)$ .

$64 x (x - 20) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 20 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 20$ equal to $0$ and solve for $x$ .

Set $x - 20$ equal to $0$ .

$x - 20 = 0$

Add $20$ to both sides of the equation.

$x = 20$

The final solution is all the values that make $64 x (x - 20) = 0$ true.

$x = 0, 20$

Step 7

Exclude the solutions that do not make $\sqrt{9 x + 16} = 8 + \sqrt{x + 16}$ true.

$x = 20$

Solve for x square root of 9x+16=8+ square root of x+16

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