Solve for x square root of x- square root of x-17=1

$\sqrt{x} - \sqrt{x - 17} = 1$

Step 1

Add $\sqrt{x - 17}$ to both sides of the equation.

$\sqrt{x} = 1 + \sqrt{x - 17}$

Step 2

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

${\sqrt{x}}^{2} = {(1 + \sqrt{x - 17})}^{2}$

Step 3

Simplify each side of the equation.

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

${(x^{\frac{1}{2}})}^{2} = {(1 + \sqrt{x - 17})}^{2}$

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

$x^{\frac{1}{2} \cdot 2} = {(1 + \sqrt{x - 17})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(1 + \sqrt{x - 17})}^{2}$

Rewrite the expression.

$x^{1} = {(1 + \sqrt{x - 17})}^{2}$

Simplify.

$x = {(1 + \sqrt{x - 17})}^{2}$

Simplify the right side.

Simplify ${(1 + \sqrt{x - 17})}^{2}$ .

Rewrite ${(1 + \sqrt{x - 17})}^{2}$ as $(1 + \sqrt{x - 17}) (1 + \sqrt{x - 17})$ .

$x = (1 + \sqrt{x - 17}) (1 + \sqrt{x - 17})$

Expand $(1 + \sqrt{x - 17}) (1 + \sqrt{x - 17})$ using the FOIL Method.

Apply the distributive property.

$x = 1 (1 + \sqrt{x - 17}) + \sqrt{x - 17} (1 + \sqrt{x - 17})$

Apply the distributive property.

$x = 1 \cdot 1 + 1 \sqrt{x - 17} + \sqrt{x - 17} (1 + \sqrt{x - 17})$

Apply the distributive property.

$x = 1 \cdot 1 + 1 \sqrt{x - 17} + \sqrt{x - 17} \cdot 1 + \sqrt{x - 17} \sqrt{x - 17}$

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

$x = 1 + 1 \sqrt{x - 17} + \sqrt{x - 17} \cdot 1 + \sqrt{x - 17} \sqrt{x - 17}$

Multiply $\sqrt{x - 17}$ by $1$ .

$x = 1 + \sqrt{x - 17} + \sqrt{x - 17} \cdot 1 + \sqrt{x - 17} \sqrt{x - 17}$

Multiply $\sqrt{x - 17}$ by $1$ .

$x = 1 + \sqrt{x - 17} + \sqrt{x - 17} + \sqrt{x - 17} \sqrt{x - 17}$

Multiply $\sqrt{x - 17} \sqrt{x - 17}$ .

Raise $\sqrt{x - 17}$ to the power of $1$ .

$x = 1 + \sqrt{x - 17} + \sqrt{x - 17} + {\sqrt{x - 17}}^{1} \sqrt{x - 17}$

Raise $\sqrt{x - 17}$ to the power of $1$ .

$x = 1 + \sqrt{x - 17} + \sqrt{x - 17} + {\sqrt{x - 17}}^{1} {\sqrt{x - 17}}^{1}$

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

$x = 1 + \sqrt{x - 17} + \sqrt{x - 17} + {\sqrt{x - 17}}^{1 + 1}$

Add $1$ and $1$ .

$x = 1 + \sqrt{x - 17} + \sqrt{x - 17} + {\sqrt{x - 17}}^{2}$

Rewrite ${\sqrt{x - 17}}^{2}$ as $x - 17$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 17}$ as ${(x - 17)}^{\frac{1}{2}}$ .

$x = 1 + \sqrt{x - 17} + \sqrt{x - 17} + {({(x - 17)}^{\frac{1}{2}})}^{2}$

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

$x = 1 + \sqrt{x - 17} + \sqrt{x - 17} + {(x - 17)}^{\frac{1}{2} \cdot 2}$

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

$x = 1 + \sqrt{x - 17} + \sqrt{x - 17} + {(x - 17)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = 1 + \sqrt{x - 17} + \sqrt{x - 17} + {(x - 17)}^{\frac{2}{2}}$

Rewrite the expression.

$x = 1 + \sqrt{x - 17} + \sqrt{x - 17} + {(x - 17)}^{1}$

Simplify.

$x = 1 + \sqrt{x - 17} + \sqrt{x - 17} + x - 17$

Subtract $17$ from $1$ .

$x = - 16 + \sqrt{x - 17} + \sqrt{x - 17} + x$

Add $\sqrt{x - 17}$ and $\sqrt{x - 17}$ .

$x = - 16 + 2 \sqrt{x - 17} + x$

Step 4

Solve for $2 \sqrt{x - 17}$ .

Rewrite the equation as $- 16 + 2 \sqrt{x - 17} + x = x$ .

$- 16 + 2 \sqrt{x - 17} + x = x$

Move all terms not containing $2 \sqrt{x - 17}$ to the right side of the equation.

Add $16$ to both sides of the equation.

$2 \sqrt{x - 17} + x = x + 16$

Subtract $x$ from both sides of the equation.

$2 \sqrt{x - 17} = x + 16 - x$

Combine the opposite terms in $x + 16 - x$ .

Subtract $x$ from $x$ .

$2 \sqrt{x - 17} = 0 + 16$

Add $0$ and $16$ .

$2 \sqrt{x - 17} = 16$

Step 5

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

${(2 \sqrt{x - 17})}^{2} = 16^{2}$

Step 6

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 17}$ as ${(x - 17)}^{\frac{1}{2}}$ .

${(2 {(x - 17)}^{\frac{1}{2}})}^{2} = 16^{2}$

Simplify the left side.

Simplify ${(2 {(x - 17)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $2 {(x - 17)}^{\frac{1}{2}}$ .

$2^{2} {({(x - 17)}^{\frac{1}{2}})}^{2} = 16^{2}$

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

$4 {({(x - 17)}^{\frac{1}{2}})}^{2} = 16^{2}$

Multiply the exponents in ${({(x - 17)}^{\frac{1}{2}})}^{2}$ .

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

$4 {(x - 17)}^{\frac{1}{2} \cdot 2} = 16^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {(x - 17)}^{\frac{1}{2} \cdot 2} = 16^{2}$

Rewrite the expression.

$4 {(x - 17)}^{1} = 16^{2}$

Simplify.

$4 (x - 17) = 16^{2}$

Apply the distributive property.

$4 x + 4 \cdot - 17 = 16^{2}$

Multiply $4$ by $- 17$ .

$4 x - 68 = 16^{2}$

Simplify the right side.

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

$4 x - 68 = 256$

Step 7

Solve for $x$ .

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

Add $68$ to both sides of the equation.

$4 x = 256 + 68$

Add $256$ and $68$ .

$4 x = 324$

Divide each term in $4 x = 324$ by $4$ and simplify.

Divide each term in $4 x = 324$ by $4$ .

$\frac{4 x}{4} = \frac{324}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{324}{4}$

Divide $x$ by $1$ .

$x = \frac{324}{4}$

Simplify the right side.

Divide $324$ by $4$ .

$x = 81$

Solve for x square root of x- square root of x-17=1

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