Solve for x square root of 7 = square root of x-1

$\sqrt{7} = \sqrt{x - 1}$

Step 1

Rewrite the equation as $\sqrt{x - 1} = \sqrt{7}$ .

$\sqrt{x - 1} = \sqrt{7}$

Step 2

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

${\sqrt{x - 1}}^{2} = {\sqrt{7}}^{2}$

Step 3

Simplify each side of the equation.

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

${({(x - 1)}^{\frac{1}{2}})}^{2} = {\sqrt{7}}^{2}$

Simplify the left side.

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

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

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

${(x - 1)}^{\frac{1}{2} \cdot 2} = {\sqrt{7}}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x - 1)}^{\frac{1}{2} \cdot 2} = {\sqrt{7}}^{2}$

Rewrite the expression.

${(x - 1)}^{1} = {\sqrt{7}}^{2}$

Simplify.

$x - 1 = {\sqrt{7}}^{2}$

Simplify the right side.

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$x - 1 = {(7^{\frac{1}{2}})}^{2}$

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

$x - 1 = 7^{\frac{1}{2} \cdot 2}$

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

$x - 1 = 7^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x - 1 = 7^{\frac{2}{2}}$

Rewrite the expression.

$x - 1 = 7^{1}$

Evaluate the exponent.

$x - 1 = 7$

Step 4

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

Add $1$ to both sides of the equation.

$x = 7 + 1$

Add $7$ and $1$ .

$x = 8$