Solve for x square root of x-8 = square root of x+4

$\sqrt{x} - 8 = \sqrt{x + 4}$

Add $8$ to both sides of the equation.

$\sqrt{x} = \sqrt{x + 4} + 8$

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

${\sqrt{x}}^{2} = {(\sqrt{x + 4} + 8)}^{2}$

Simplify each side of the equation.

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

${(x^{\frac{1}{2}})}^{2} = {(\sqrt{x + 4} + 8)}^{2}$

Simplify the left side.

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

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

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

$x^{\frac{1}{2} \cdot 2} = {(\sqrt{x + 4} + 8)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(\sqrt{x + 4} + 8)}^{2}$

Rewrite the expression.

$x^{1} = {(\sqrt{x + 4} + 8)}^{2}$

Simplify.

$x = {(\sqrt{x + 4} + 8)}^{2}$

Simplify the right side.

Simplify ${(\sqrt{x + 4} + 8)}^{2}$ .

Rewrite ${(\sqrt{x + 4} + 8)}^{2}$ as $(\sqrt{x + 4} + 8) (\sqrt{x + 4} + 8)$ .

$x = (\sqrt{x + 4} + 8) (\sqrt{x + 4} + 8)$

Expand $(\sqrt{x + 4} + 8) (\sqrt{x + 4} + 8)$ using the FOIL Method.

Apply the distributive property.

$x = \sqrt{x + 4} (\sqrt{x + 4} + 8) + 8 (\sqrt{x + 4} + 8)$

Apply the distributive property.

$x = \sqrt{x + 4} \sqrt{x + 4} + \sqrt{x + 4} \cdot 8 + 8 (\sqrt{x + 4} + 8)$

Apply the distributive property.

$x = \sqrt{x + 4} \sqrt{x + 4} + \sqrt{x + 4} \cdot 8 + 8 \sqrt{x + 4} + 8 \cdot 8$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{x + 4} \sqrt{x + 4}$ .

Raise $\sqrt{x + 4}$ to the power of $1$ .

$x = {\sqrt{x + 4}}^{1} \sqrt{x + 4} + \sqrt{x + 4} \cdot 8 + 8 \sqrt{x + 4} + 8 \cdot 8$

Raise $\sqrt{x + 4}$ to the power of $1$ .

$x = {\sqrt{x + 4}}^{1} {\sqrt{x + 4}}^{1} + \sqrt{x + 4} \cdot 8 + 8 \sqrt{x + 4} + 8 \cdot 8$

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

$x = {\sqrt{x + 4}}^{1 + 1} + \sqrt{x + 4} \cdot 8 + 8 \sqrt{x + 4} + 8 \cdot 8$

Add $1$ and $1$ .

$x = {\sqrt{x + 4}}^{2} + \sqrt{x + 4} \cdot 8 + 8 \sqrt{x + 4} + 8 \cdot 8$

Rewrite ${\sqrt{x + 4}}^{2}$ as $x + 4$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 4}$ as ${(x + 4)}^{\frac{1}{2}}$ .

$x = {({(x + 4)}^{\frac{1}{2}})}^{2} + \sqrt{x + 4} \cdot 8 + 8 \sqrt{x + 4} + 8 \cdot 8$

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

$x = {(x + 4)}^{\frac{1}{2} \cdot 2} + \sqrt{x + 4} \cdot 8 + 8 \sqrt{x + 4} + 8 \cdot 8$

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

$x = {(x + 4)}^{\frac{2}{2}} + \sqrt{x + 4} \cdot 8 + 8 \sqrt{x + 4} + 8 \cdot 8$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = {(x + 4)}^{\frac{2}{2}} + \sqrt{x + 4} \cdot 8 + 8 \sqrt{x + 4} + 8 \cdot 8$

Rewrite the expression.

$x = {(x + 4)}^{1} + \sqrt{x + 4} \cdot 8 + 8 \sqrt{x + 4} + 8 \cdot 8$

Simplify.

$x = x + 4 + \sqrt{x + 4} \cdot 8 + 8 \sqrt{x + 4} + 8 \cdot 8$

Move $8$ to the left of $\sqrt{x + 4}$ .

$x = x + 4 + 8 \cdot \sqrt{x + 4} + 8 \sqrt{x + 4} + 8 \cdot 8$

Multiply $8$ by $8$ .

$x = x + 4 + 8 \sqrt{x + 4} + 8 \sqrt{x + 4} + 64$

Add $4$ and $64$ .

$x = x + 68 + 8 \sqrt{x + 4} + 8 \sqrt{x + 4}$

Add $8 \sqrt{x + 4}$ and $8 \sqrt{x + 4}$ .

$x = x + 68 + 16 \sqrt{x + 4}$

Solve for $16 \sqrt{x + 4}$ .

Rewrite the equation as $x + 68 + 16 \sqrt{x + 4} = x$ .

$x + 68 + 16 \sqrt{x + 4} = x$

Move all terms not containing $16 \sqrt{x + 4}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

$68 + 16 \sqrt{x + 4} = x - x$

Subtract $68$ from both sides of the equation.

$16 \sqrt{x + 4} = x - x - 68$

Subtract $x$ from $x$ .

$16 \sqrt{x + 4} = 0 - 68$

Subtract $68$ from $0$ .

$16 \sqrt{x + 4} = - 68$

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

${(16 \sqrt{x + 4})}^{2} = {(- 68)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 4}$ as ${(x + 4)}^{\frac{1}{2}}$ .

${(16 {(x + 4)}^{\frac{1}{2}})}^{2} = {(- 68)}^{2}$

Simplify the left side.

Simplify ${(16 {(x + 4)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $16 {(x + 4)}^{\frac{1}{2}}$ .

$16^{2} {({(x + 4)}^{\frac{1}{2}})}^{2} = {(- 68)}^{2}$

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

$256 {({(x + 4)}^{\frac{1}{2}})}^{2} = {(- 68)}^{2}$

Multiply the exponents in ${({(x + 4)}^{\frac{1}{2}})}^{2}$ .

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

$256 {(x + 4)}^{\frac{1}{2} \cdot 2} = {(- 68)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$256 {(x + 4)}^{\frac{1}{2} \cdot 2} = {(- 68)}^{2}$

Rewrite the expression.

$256 {(x + 4)}^{1} = {(- 68)}^{2}$

Simplify.

$256 (x + 4) = {(- 68)}^{2}$

Apply the distributive property.

$256 x + 256 \cdot 4 = {(- 68)}^{2}$

Multiply $256$ by $4$ .

$256 x + 1024 = {(- 68)}^{2}$

Simplify the right side.

Raise $- 68$ to the power of $2$ .

$256 x + 1024 = 4624$

Solve for $x$ .

Move all terms not containing $x$ to the right side of the equation.

Subtract $1024$ from both sides of the equation.

$256 x = 4624 - 1024$

Subtract $1024$ from $4624$ .

$256 x = 3600$

Divide each term in $256 x = 3600$ by $256$ and simplify.

Divide each term in $256 x = 3600$ by $256$ .

$\frac{256 x}{256} = \frac{3600}{256}$

Simplify the left side.

Cancel the common factor of $256$ .

Cancel the common factor.

$\frac{256 x}{256} = \frac{3600}{256}$

Divide $x$ by $1$ .

$x = \frac{3600}{256}$

Simplify the right side.

Cancel the common factor of $3600$ and $256$ .

Factor $16$ out of $3600$ .

$x = \frac{16 (225)}{256}$

Cancel the common factors.

Factor $16$ out of $256$ .

$x = \frac{16 \cdot 225}{16 \cdot 16}$

Cancel the common factor.

$x = \frac{16 \cdot 225}{16 \cdot 16}$

Rewrite the expression.

$x = \frac{225}{16}$

Exclude the solutions that do not make $\sqrt{x} - 8 = \sqrt{x + 4}$ true.

$No solution$

Solve for x square root of x-8 = square root of x+4

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