Solve for x square root of x- square root of x-16=4

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

Step 1

Add $\sqrt{x - 16}$ to both sides of the equation.

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

Step 2

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

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

Step 3

Simplify each side of the equation.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

Simplify ${(4 + \sqrt{x - 16})}^{2}$ .

Rewrite ${(4 + \sqrt{x - 16})}^{2}$ as $(4 + \sqrt{x - 16}) (4 + \sqrt{x - 16})$ .

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

Expand $(4 + \sqrt{x - 16}) (4 + \sqrt{x - 16})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

$x = 4 \cdot 4 + 4 \sqrt{x - 16} + \sqrt{x - 16} (4 + \sqrt{x - 16})$

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $4$ .

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

Move $4$ to the left of $\sqrt{x - 16}$ .

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

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

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

$x = 16 + 4 \sqrt{x - 16} + 4 \sqrt{x - 16} + {\sqrt{x - 16}}^{1} \sqrt{x - 16}$

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

$x = 16 + 4 \sqrt{x - 16} + 4 \sqrt{x - 16} + {\sqrt{x - 16}}^{1} {\sqrt{x - 16}}^{1}$

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

$x = 16 + 4 \sqrt{x - 16} + 4 \sqrt{x - 16} + {\sqrt{x - 16}}^{1 + 1}$

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$x = 16 + 4 \sqrt{x - 16} + 4 \sqrt{x - 16} + {(x - 16)}^{1}$

Simplify.

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

Subtract $16$ from $16$ .

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

Add $0$ and $4 \sqrt{x - 16}$ .

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

Add $4 \sqrt{x - 16}$ and $4 \sqrt{x - 16}$ .

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

Step 4

Solve for $8 \sqrt{x - 16}$ .

Rewrite the equation as $8 \sqrt{x - 16} + x = x$ .

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

Move all terms not containing $8 \sqrt{x - 16}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

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

Subtract $x$ from $x$ .

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

Step 5

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

${(8 \sqrt{x - 16})}^{2} = 0^{2}$

Step 6

Simplify each side of the equation.

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

${(8 {(x - 16)}^{\frac{1}{2}})}^{2} = 0^{2}$

Simplify the left side.

Simplify ${(8 {(x - 16)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $8 {(x - 16)}^{\frac{1}{2}}$ .

$8^{2} {({(x - 16)}^{\frac{1}{2}})}^{2} = 0^{2}$

Raise $8$ to the power of $2$ .

$64 {({(x - 16)}^{\frac{1}{2}})}^{2} = 0^{2}$

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

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

$64 {(x - 16)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$64 {(x - 16)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Rewrite the expression.

$64 {(x - 16)}^{1} = 0^{2}$

Simplify.

$64 (x - 16) = 0^{2}$

Apply the distributive property.

$64 x + 64 \cdot - 16 = 0^{2}$

Multiply $64$ by $- 16$ .

$64 x - 1024 = 0^{2}$

Simplify the right side.

Raising $0$ to any positive power yields $0$ .

$64 x - 1024 = 0$

Step 7

Solve for $x$ .

Add $1024$ to both sides of the equation.

$64 x = 1024$

Divide each term in $64 x = 1024$ by $64$ and simplify.

Divide each term in $64 x = 1024$ by $64$ .

$\frac{64 x}{64} = \frac{1024}{64}$

Simplify the left side.

Cancel the common factor of $64$ .

Cancel the common factor.

$\frac{64 x}{64} = \frac{1024}{64}$

Divide $x$ by $1$ .

$x = \frac{1024}{64}$

Simplify the right side.

Divide $1024$ by $64$ .

$x = 16$

Solve for x square root of x- square root of x-16=4

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