Solve for k 1+ square root of k+1=9

$1 + \sqrt{k + 1} = 9$

Step 1

Move all terms not containing $\sqrt{k + 1}$ to the right side of the equation.

Subtract $1$ from both sides of the equation.

$\sqrt{k + 1} = 9 - 1$

Subtract $1$ from $9$ .

$\sqrt{k + 1} = 8$

Step 2

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

${\sqrt{k + 1}}^{2} = 8^{2}$

Step 3

Simplify each side of the equation.

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

${({(k + 1)}^{\frac{1}{2}})}^{2} = 8^{2}$

Simplify the left side.

Simplify ${({(k + 1)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(k + 1)}^{\frac{1}{2}})}^{2}$ .

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

${(k + 1)}^{\frac{1}{2} \cdot 2} = 8^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(k + 1)}^{\frac{1}{2} \cdot 2} = 8^{2}$

Rewrite the expression.

${(k + 1)}^{1} = 8^{2}$

Simplify.

$k + 1 = 8^{2}$

Simplify the right side.

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

$k + 1 = 64$

Step 4

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

Subtract $1$ from both sides of the equation.

$k = 64 - 1$

Subtract $1$ from $64$ .

$k = 63$