Solve for x square root of 3+ square root of x=3

$\sqrt{3 + \sqrt{x}} = 3$

Step 1

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

${\sqrt{3 + \sqrt{x}}}^{2} = 3^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 + \sqrt{x}}$ as ${(3 + \sqrt{x})}^{\frac{1}{2}}$ .

${({(3 + \sqrt{x})}^{\frac{1}{2}})}^{2} = 3^{2}$

Simplify the left side.

Simplify ${({(3 + \sqrt{x})}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(3 + \sqrt{x})}^{\frac{1}{2}})}^{2}$ .

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

${(3 + \sqrt{x})}^{\frac{1}{2} \cdot 2} = 3^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(3 + \sqrt{x})}^{\frac{1}{2} \cdot 2} = 3^{2}$

Rewrite the expression.

${(3 + \sqrt{x})}^{1} = 3^{2}$

Simplify.

$3 + \sqrt{x} = 3^{2}$

Simplify the right side.

Raise $3$ to the power of $2$ .

$3 + \sqrt{x} = 9$

Step 3

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

Subtract $3$ from both sides of the equation.

$\sqrt{x} = 9 - 3$

Subtract $3$ from $9$ .

$\sqrt{x} = 6$

Step 4

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

${\sqrt{x}}^{2} = 6^{2}$

Step 5

Simplify each side of the equation.

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

${(x^{\frac{1}{2}})}^{2} = 6^{2}$

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

$x^{\frac{1}{2} \cdot 2} = 6^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 6^{2}$

Rewrite the expression.

$x^{1} = 6^{2}$

Simplify.

$x = 6^{2}$

Simplify the right side.

Raise $6$ to the power of $2$ .

$x = 36$

Solve for x square root of 3+ square root of x=3

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