Solve for x square root of x- square root of x-32=2

$\sqrt{x} - \sqrt{x - 32} = 2$

Step 1

Add $\sqrt{x - 32}$ to both sides of the equation.

$\sqrt{x} = 2 + \sqrt{x - 32}$

Step 2

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

${\sqrt{x}}^{2} = {(2 + \sqrt{x - 32})}^{2}$

Step 3

Simplify each side of the equation.

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

${(x^{\frac{1}{2}})}^{2} = {(2 + \sqrt{x - 32})}^{2}$

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

$x^{\frac{1}{2} \cdot 2} = {(2 + \sqrt{x - 32})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(2 + \sqrt{x - 32})}^{2}$

Rewrite the expression.

$x^{1} = {(2 + \sqrt{x - 32})}^{2}$

Simplify.

$x = {(2 + \sqrt{x - 32})}^{2}$

Simplify the right side.

Simplify ${(2 + \sqrt{x - 32})}^{2}$ .

Rewrite ${(2 + \sqrt{x - 32})}^{2}$ as $(2 + \sqrt{x - 32}) (2 + \sqrt{x - 32})$ .

$x = (2 + \sqrt{x - 32}) (2 + \sqrt{x - 32})$

Expand $(2 + \sqrt{x - 32}) (2 + \sqrt{x - 32})$ using the FOIL Method.

Apply the distributive property.

$x = 2 (2 + \sqrt{x - 32}) + \sqrt{x - 32} (2 + \sqrt{x - 32})$

Apply the distributive property.

$x = 2 \cdot 2 + 2 \sqrt{x - 32} + \sqrt{x - 32} (2 + \sqrt{x - 32})$

Apply the distributive property.

$x = 2 \cdot 2 + 2 \sqrt{x - 32} + \sqrt{x - 32} \cdot 2 + \sqrt{x - 32} \sqrt{x - 32}$

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

$x = 4 + 2 \sqrt{x - 32} + \sqrt{x - 32} \cdot 2 + \sqrt{x - 32} \sqrt{x - 32}$

Move $2$ to the left of $\sqrt{x - 32}$ .

$x = 4 + 2 \sqrt{x - 32} + 2 \cdot \sqrt{x - 32} + \sqrt{x - 32} \sqrt{x - 32}$

Multiply $\sqrt{x - 32} \sqrt{x - 32}$ .

Raise $\sqrt{x - 32}$ to the power of $1$ .

$x = 4 + 2 \sqrt{x - 32} + 2 \sqrt{x - 32} + {\sqrt{x - 32}}^{1} \sqrt{x - 32}$

Raise $\sqrt{x - 32}$ to the power of $1$ .

$x = 4 + 2 \sqrt{x - 32} + 2 \sqrt{x - 32} + {\sqrt{x - 32}}^{1} {\sqrt{x - 32}}^{1}$

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

$x = 4 + 2 \sqrt{x - 32} + 2 \sqrt{x - 32} + {\sqrt{x - 32}}^{1 + 1}$

Add $1$ and $1$ .

$x = 4 + 2 \sqrt{x - 32} + 2 \sqrt{x - 32} + {\sqrt{x - 32}}^{2}$

Rewrite ${\sqrt{x - 32}}^{2}$ as $x - 32$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 32}$ as ${(x - 32)}^{\frac{1}{2}}$ .

$x = 4 + 2 \sqrt{x - 32} + 2 \sqrt{x - 32} + {({(x - 32)}^{\frac{1}{2}})}^{2}$

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

$x = 4 + 2 \sqrt{x - 32} + 2 \sqrt{x - 32} + {(x - 32)}^{\frac{1}{2} \cdot 2}$

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

$x = 4 + 2 \sqrt{x - 32} + 2 \sqrt{x - 32} + {(x - 32)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = 4 + 2 \sqrt{x - 32} + 2 \sqrt{x - 32} + {(x - 32)}^{\frac{2}{2}}$

Rewrite the expression.

$x = 4 + 2 \sqrt{x - 32} + 2 \sqrt{x - 32} + {(x - 32)}^{1}$

Simplify.

$x = 4 + 2 \sqrt{x - 32} + 2 \sqrt{x - 32} + x - 32$

Subtract $32$ from $4$ .

$x = - 28 + 2 \sqrt{x - 32} + 2 \sqrt{x - 32} + x$

Add $2 \sqrt{x - 32}$ and $2 \sqrt{x - 32}$ .

$x = - 28 + 4 \sqrt{x - 32} + x$

Step 4

Solve for $4 \sqrt{x - 32}$ .

Rewrite the equation as $- 28 + 4 \sqrt{x - 32} + x = x$ .

$- 28 + 4 \sqrt{x - 32} + x = x$

Move all terms not containing $4 \sqrt{x - 32}$ to the right side of the equation.

Add $28$ to both sides of the equation.

$4 \sqrt{x - 32} + x = x + 28$

Subtract $x$ from both sides of the equation.

$4 \sqrt{x - 32} = x + 28 - x$

Combine the opposite terms in $x + 28 - x$ .

Subtract $x$ from $x$ .

$4 \sqrt{x - 32} = 0 + 28$

Add $0$ and $28$ .

$4 \sqrt{x - 32} = 28$

Step 5

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

${(4 \sqrt{x - 32})}^{2} = 28^{2}$

Step 6

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 32}$ as ${(x - 32)}^{\frac{1}{2}}$ .

${(4 {(x - 32)}^{\frac{1}{2}})}^{2} = 28^{2}$

Simplify the left side.

Simplify ${(4 {(x - 32)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $4 {(x - 32)}^{\frac{1}{2}}$ .

$4^{2} {({(x - 32)}^{\frac{1}{2}})}^{2} = 28^{2}$

Raise $4$ to the power of $2$ .

$16 {({(x - 32)}^{\frac{1}{2}})}^{2} = 28^{2}$

Multiply the exponents in ${({(x - 32)}^{\frac{1}{2}})}^{2}$ .

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

$16 {(x - 32)}^{\frac{1}{2} \cdot 2} = 28^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$16 {(x - 32)}^{\frac{1}{2} \cdot 2} = 28^{2}$

Rewrite the expression.

$16 {(x - 32)}^{1} = 28^{2}$

Simplify.

$16 (x - 32) = 28^{2}$

Apply the distributive property.

$16 x + 16 \cdot - 32 = 28^{2}$

Multiply $16$ by $- 32$ .

$16 x - 512 = 28^{2}$

Simplify the right side.

Raise $28$ to the power of $2$ .

$16 x - 512 = 784$

Step 7

Solve for $x$ .

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

Add $512$ to both sides of the equation.

$16 x = 784 + 512$

Add $784$ and $512$ .

$16 x = 1296$

Divide each term in $16 x = 1296$ by $16$ and simplify.

Divide each term in $16 x = 1296$ by $16$ .

$\frac{16 x}{16} = \frac{1296}{16}$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

$\frac{16 x}{16} = \frac{1296}{16}$

Divide $x$ by $1$ .

$x = \frac{1296}{16}$

Simplify the right side.

Divide $1296$ by $16$ .

$x = 81$

Solve for x square root of x- square root of x-32=2

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