Solve for x square root of 7+ square root of 2x=3

$\sqrt{7 + \sqrt{2 x}} = 3$

Step 1

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

${\sqrt{7 + \sqrt{2 x}}}^{2} = 3^{2}$

Step 2

Simplify each side of the equation.

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

${({(7 + \sqrt{2 x})}^{\frac{1}{2}})}^{2} = 3^{2}$

Simplify the left side.

Simplify ${({(7 + \sqrt{2 x})}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(7 + \sqrt{2 x})}^{\frac{1}{2}})}^{2}$ .

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

${(7 + \sqrt{2 x})}^{\frac{1}{2} \cdot 2} = 3^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(7 + \sqrt{2 x})}^{\frac{1}{2} \cdot 2} = 3^{2}$

Rewrite the expression.

${(7 + \sqrt{2 x})}^{1} = 3^{2}$

Simplify.

$7 + \sqrt{2 x} = 3^{2}$

Simplify the right side.

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

$7 + \sqrt{2 x} = 9$

Step 3

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

Subtract $7$ from both sides of the equation.

$\sqrt{2 x} = 9 - 7$

Subtract $7$ from $9$ .

$\sqrt{2 x} = 2$

Step 4

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

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

Step 5

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

${(2 x)}^{1} = 2^{2}$

Simplify.

$2 x = 2^{2}$

Simplify the right side.

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

$2 x = 4$

Step 6

Divide each term in $2 x = 4$ by $2$ and simplify.

Divide each term in $2 x = 4$ by $2$ .

$\frac{2 x}{2} = \frac{4}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{4}{2}$

Divide $x$ by $1$ .

$x = \frac{4}{2}$

Simplify the right side.

Divide $4$ by $2$ .

$x = 2$

Solve for x square root of 7+ square root of 2x=3

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