Solve for x square root of square root of x+9 = square root of x+3

$\sqrt{\sqrt{x + 9}} = \sqrt{x + 3}$

Step 1

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

${\sqrt{\sqrt{x + 9}}}^{2} = {\sqrt{x + 3}}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\sqrt{x + 9}}$ as ${\sqrt{x + 9}}^{\frac{1}{2}}$ .

${({\sqrt{x + 9}}^{\frac{1}{2}})}^{2} = {\sqrt{x + 3}}^{2}$

Simplify the left side.

Simplify ${({\sqrt{x + 9}}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({\sqrt{x + 9}}^{\frac{1}{2}})}^{2}$ .

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

${\sqrt{x + 9}}^{\frac{1}{2} \cdot 2} = {\sqrt{x + 3}}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${\sqrt{x + 9}}^{\frac{1}{2} \cdot 2} = {\sqrt{x + 3}}^{2}$

Rewrite the expression.

${\sqrt{x + 9}}^{1} = {\sqrt{x + 3}}^{2}$

Simplify.

$\sqrt{x + 9} = {\sqrt{x + 3}}^{2}$

Simplify the right side.

Rewrite ${\sqrt{x + 3}}^{2}$ as $x + 3$ .

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

$\sqrt{x + 9} = {({(x + 3)}^{\frac{1}{2}})}^{2}$

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

$\sqrt{x + 9} = {(x + 3)}^{\frac{1}{2} \cdot 2}$

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

$\sqrt{x + 9} = {(x + 3)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{x + 9} = {(x + 3)}^{\frac{2}{2}}$

Rewrite the expression.

$\sqrt{x + 9} = {(x + 3)}^{1}$

Simplify.

$\sqrt{x + 9} = x + 3$

Step 3

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

${\sqrt{x + 9}}^{2} = {(x + 3)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 9}$ as ${(x + 9)}^{\frac{1}{2}}$ .

${({(x + 9)}^{\frac{1}{2}})}^{2} = {(x + 3)}^{2}$

Simplify the left side.

Simplify ${({(x + 9)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x + 9)}^{\frac{1}{2}})}^{2}$ .

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

${(x + 9)}^{\frac{1}{2} \cdot 2} = {(x + 3)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x + 9)}^{\frac{1}{2} \cdot 2} = {(x + 3)}^{2}$

Rewrite the expression.

${(x + 9)}^{1} = {(x + 3)}^{2}$

Simplify.

$x + 9 = {(x + 3)}^{2}$

Simplify the right side.

Simplify ${(x + 3)}^{2}$ .

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$x + 9 = (x + 3) (x + 3)$

Expand $(x + 3) (x + 3)$ using the FOIL Method.

Apply the distributive property.

$x + 9 = x (x + 3) + 3 (x + 3)$

Apply the distributive property.

$x + 9 = x \cdot x + x \cdot 3 + 3 (x + 3)$

Apply the distributive property.

$x + 9 = x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x + 9 = x^{2} + x \cdot 3 + 3 x + 3 \cdot 3$

Move $3$ to the left of $x$ .

$x + 9 = x^{2} + 3 \cdot x + 3 x + 3 \cdot 3$

Multiply $3$ by $3$ .

$x + 9 = x^{2} + 3 x + 3 x + 9$

Add $3 x$ and $3 x$ .

$x + 9 = x^{2} + 6 x + 9$

Step 5

Solve for $x$ .

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} + 6 x + 9 = x + 9$

Move all terms containing $x$ to the left side of the equation.

Subtract $x$ from both sides of the equation.

$x^{2} + 6 x + 9 - x = 9$

Subtract $x$ from $6 x$ .

$x^{2} + 5 x + 9 = 9$

Subtract $9$ from both sides of the equation.

$x^{2} + 5 x + 9 - 9 = 0$

Combine the opposite terms in $x^{2} + 5 x + 9 - 9$ .

Subtract $9$ from $9$ .

$x^{2} + 5 x + 0 = 0$

Add $x^{2} + 5 x$ and $0$ .

$x^{2} + 5 x = 0$

Factor $x$ out of $x^{2} + 5 x$ .

Factor $x$ out of $x^{2}$ .

$x \cdot x + 5 x = 0$

Factor $x$ out of $5 x$ .

$x \cdot x + x \cdot 5 = 0$

Factor $x$ out of $x \cdot x + x \cdot 5$ .

$x (x + 5) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x + 5 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x + 5$ equal to $0$ and solve for $x$ .

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

The final solution is all the values that make $x (x + 5) = 0$ true.

$x = 0, - 5$

Step 6

Exclude the solutions that do not make $\sqrt{\sqrt{x + 9}} = \sqrt{x + 3}$ true.

$x = 0$

Solve for x square root of square root of x+9 = square root of x+3

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