Solve for x 2 square root of x-1 = square root of 32

$2 \sqrt{x - 1} = \sqrt{32}$

Step 1

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

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

Step 2

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Apply the distributive property.

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

Multiply $4$ by $- 1$ .

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

Simplify the right side.

Simplify ${\sqrt{32}}^{2}$ .

Rewrite $32$ as $4^{2} \cdot 2$ .

Factor $16$ out of $32$ .

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

Rewrite $16$ as $4^{2}$ .

$4 x - 4 = {\sqrt{4^{2} \cdot 2}}^{2}$

Pull terms out from under the radical.

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

Simplify the expression.

Apply the product rule to $4 \sqrt{2}$ .

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

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

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

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$4 x - 4 = 16 \cdot 2^{1}$

Evaluate the exponent.

$4 x - 4 = 16 \cdot 2$

Multiply $16$ by $2$ .

$4 x - 4 = 32$

Step 3

Solve for $x$ .

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

Add $4$ to both sides of the equation.

$4 x = 32 + 4$

Add $32$ and $4$ .

$4 x = 36$

Divide each term in $4 x = 36$ by $4$ and simplify.

Divide each term in $4 x = 36$ by $4$ .

$\frac{4 x}{4} = \frac{36}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{36}{4}$

Divide $x$ by $1$ .

$x = \frac{36}{4}$

Simplify the right side.

Divide $36$ by $4$ .

$x = 9$

Solve for x 2 square root of x-1 = square root of 32

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