Solve by Factoring square root of x+ square root of x-16=8

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

Step 1

Subtract $8$ from both sides of the equation.

$\sqrt{x} + \sqrt{x - 16} - 8 = 0$

Step 2

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

Subtract $\sqrt{x - 16}$ from both sides of the equation.

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

Add $8$ to both sides of the equation.

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

Step 3

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

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

Step 4

Simplify each side of the equation.

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

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $- \sqrt{x - 16} (- \sqrt{x - 16})$ .

Multiply $- 1$ by $- 1$ .

$x = 1 \sqrt{x - 16} \sqrt{x - 16} - \sqrt{x - 16} \cdot 8 + 8 (- \sqrt{x - 16}) + 8 \cdot 8$

Multiply $\sqrt{x - 16}$ by $1$ .

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

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

$x = {\sqrt{x - 16}}^{1} \sqrt{x - 16} - \sqrt{x - 16} \cdot 8 + 8 (- \sqrt{x - 16}) + 8 \cdot 8$

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

$x = {\sqrt{x - 16}}^{1} {\sqrt{x - 16}}^{1} - \sqrt{x - 16} \cdot 8 + 8 (- \sqrt{x - 16}) + 8 \cdot 8$

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

$x = {\sqrt{x - 16}}^{1 + 1} - \sqrt{x - 16} \cdot 8 + 8 (- \sqrt{x - 16}) + 8 \cdot 8$

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$x = {(x - 16)}^{1} - \sqrt{x - 16} \cdot 8 + 8 (- \sqrt{x - 16}) + 8 \cdot 8$

Simplify.

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

Multiply $8$ by $- 1$ .

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

Multiply $- 1$ by $8$ .

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

Multiply $8$ by $8$ .

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

Add $- 16$ and $64$ .

$x = x + 48 - 8 \sqrt{x - 16} - 8 \sqrt{x - 16}$

Subtract $8 \sqrt{x - 16}$ from $- 8 \sqrt{x - 16}$ .

$x = x + 48 - 16 \sqrt{x - 16}$

Step 5

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

Rewrite the equation as $x + 48 - 16 \sqrt{x - 16} = x$ .

$x + 48 - 16 \sqrt{x - 16} = x$

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

Subtract $x$ from both sides of the equation.

$48 - 16 \sqrt{x - 16} = x - x$

Subtract $48$ from both sides of the equation.

$- 16 \sqrt{x - 16} = x - x - 48$

Subtract $x$ from $x$ .

$- 16 \sqrt{x - 16} = 0 - 48$

Subtract $48$ from $0$ .

$- 16 \sqrt{x - 16} = - 48$

Step 6

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

${(- 16 \sqrt{x - 16})}^{2} = {(- 48)}^{2}$

Step 7

Simplify each side of the equation.

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

${(- 16 {(x - 16)}^{\frac{1}{2}})}^{2} = {(- 48)}^{2}$

Simplify the left side.

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

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

${(- 16)}^{2} {({(x - 16)}^{\frac{1}{2}})}^{2} = {(- 48)}^{2}$

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

$256 {({(x - 16)}^{\frac{1}{2}})}^{2} = {(- 48)}^{2}$

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

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

$256 {(x - 16)}^{\frac{1}{2} \cdot 2} = {(- 48)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$256 {(x - 16)}^{\frac{1}{2} \cdot 2} = {(- 48)}^{2}$

Rewrite the expression.

$256 {(x - 16)}^{1} = {(- 48)}^{2}$

Simplify.

$256 (x - 16) = {(- 48)}^{2}$

Apply the distributive property.

$256 x + 256 \cdot - 16 = {(- 48)}^{2}$

Multiply $256$ by $- 16$ .

$256 x - 4096 = {(- 48)}^{2}$

Simplify the right side.

Raise $- 48$ to the power of $2$ .

$256 x - 4096 = 2304$

Step 8

Solve for $x$ .

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

Add $4096$ to both sides of the equation.

$256 x = 2304 + 4096$

Add $2304$ and $4096$ .

$256 x = 6400$

Divide each term in $256 x = 6400$ by $256$ and simplify.

Divide each term in $256 x = 6400$ by $256$ .

$\frac{256 x}{256} = \frac{6400}{256}$

Simplify the left side.

Cancel the common factor of $256$ .

Cancel the common factor.

$\frac{256 x}{256} = \frac{6400}{256}$

Divide $x$ by $1$ .

$x = \frac{6400}{256}$

Simplify the right side.

Divide $6400$ by $256$ .

$x = 25$

Solve by Factoring square root of x+ square root of x-16=8

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