Solve for r square root of r+12=2+ square root of r

$\sqrt{r + 12} = 2 + \sqrt{r}$

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

${\sqrt{r + 12}}^{2} = {(2 + \sqrt{r})}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{r + 12}$ as ${(r + 12)}^{\frac{1}{2}}$ .

${({(r + 12)}^{\frac{1}{2}})}^{2} = {(2 + \sqrt{r})}^{2}$

Simplify the left side.

Simplify ${({(r + 12)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(r + 12)}^{\frac{1}{2}})}^{2}$ .

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

${(r + 12)}^{\frac{1}{2} \cdot 2} = {(2 + \sqrt{r})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(r + 12)}^{\frac{1}{2} \cdot 2} = {(2 + \sqrt{r})}^{2}$

Rewrite the expression.

${(r + 12)}^{1} = {(2 + \sqrt{r})}^{2}$

Simplify.

$r + 12 = {(2 + \sqrt{r})}^{2}$

Simplify the right side.

Simplify ${(2 + \sqrt{r})}^{2}$ .

Rewrite ${(2 + \sqrt{r})}^{2}$ as $(2 + \sqrt{r}) (2 + \sqrt{r})$ .

$r + 12 = (2 + \sqrt{r}) (2 + \sqrt{r})$

Expand $(2 + \sqrt{r}) (2 + \sqrt{r})$ using the FOIL Method.

Apply the distributive property.

$r + 12 = 2 (2 + \sqrt{r}) + \sqrt{r} (2 + \sqrt{r})$

Apply the distributive property.

$r + 12 = 2 \cdot 2 + 2 \sqrt{r} + \sqrt{r} (2 + \sqrt{r})$

Apply the distributive property.

$r + 12 = 2 \cdot 2 + 2 \sqrt{r} + \sqrt{r} \cdot 2 + \sqrt{r} \sqrt{r}$

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

$r + 12 = 4 + 2 \sqrt{r} + \sqrt{r} \cdot 2 + \sqrt{r} \sqrt{r}$

Move $2$ to the left of $\sqrt{r}$ .

$r + 12 = 4 + 2 \sqrt{r} + 2 \cdot \sqrt{r} + \sqrt{r} \sqrt{r}$

Multiply $\sqrt{r} \sqrt{r}$ .

Raise $\sqrt{r}$ to the power of $1$ .

$r + 12 = 4 + 2 \sqrt{r} + 2 \sqrt{r} + {\sqrt{r}}^{1} \sqrt{r}$

Raise $\sqrt{r}$ to the power of $1$ .

$r + 12 = 4 + 2 \sqrt{r} + 2 \sqrt{r} + {\sqrt{r}}^{1} {\sqrt{r}}^{1}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$r + 12 = 4 + 2 \sqrt{r} + 2 \sqrt{r} + {\sqrt{r}}^{1 + 1}$

Add $1$ and $1$ .

$r + 12 = 4 + 2 \sqrt{r} + 2 \sqrt{r} + {\sqrt{r}}^{2}$

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

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

$r + 12 = 4 + 2 \sqrt{r} + 2 \sqrt{r} + {(r^{\frac{1}{2}})}^{2}$

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

$r + 12 = 4 + 2 \sqrt{r} + 2 \sqrt{r} + r^{\frac{1}{2} \cdot 2}$

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

$r + 12 = 4 + 2 \sqrt{r} + 2 \sqrt{r} + r^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$r + 12 = 4 + 2 \sqrt{r} + 2 \sqrt{r} + r^{\frac{2}{2}}$

Rewrite the expression.

$r + 12 = 4 + 2 \sqrt{r} + 2 \sqrt{r} + r^{1}$

Simplify.

$r + 12 = 4 + 2 \sqrt{r} + 2 \sqrt{r} + r$

Add $2 \sqrt{r}$ and $2 \sqrt{r}$ .

$r + 12 = 4 + 4 \sqrt{r} + r$

Solve for $4 \sqrt{r}$ .

Rewrite the equation as $4 + 4 \sqrt{r} + r = r + 12$ .

$4 + 4 \sqrt{r} + r = r + 12$

Move all terms not containing $4 \sqrt{r}$ to the right side of the equation.

Subtract $4$ from both sides of the equation.

$4 \sqrt{r} + r = r + 12 - 4$

Subtract $r$ from both sides of the equation.

$4 \sqrt{r} = r + 12 - 4 - r$

Combine the opposite terms in $r + 12 - 4 - r$ .

Subtract $r$ from $r$ .

$4 \sqrt{r} = 0 + 12 - 4$

Add $0$ and $12$ .

$4 \sqrt{r} = 12 - 4$

Subtract $4$ from $12$ .

$4 \sqrt{r} = 8$

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

${(4 \sqrt{r})}^{2} = 8^{2}$

Simplify each side of the equation.

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

${(4 r^{\frac{1}{2}})}^{2} = 8^{2}$

Simplify the left side.

Simplify ${(4 r^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $4 r^{\frac{1}{2}}$ .

$4^{2} {(r^{\frac{1}{2}})}^{2} = 8^{2}$

Raise $4$ to the power of $2$ .

$16 {(r^{\frac{1}{2}})}^{2} = 8^{2}$

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

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

$16 r^{\frac{1}{2} \cdot 2} = 8^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$16 r^{\frac{1}{2} \cdot 2} = 8^{2}$

Rewrite the expression.

$16 r^{1} = 8^{2}$

Simplify.

$16 r = 8^{2}$

Simplify the right side.

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

$16 r = 64$

Divide each term in $16 r = 64$ by $16$ and simplify.

Divide each term in $16 r = 64$ by $16$ .

$\frac{16 r}{16} = \frac{64}{16}$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

$\frac{16 r}{16} = \frac{64}{16}$

Divide $r$ by $1$ .

$r = \frac{64}{16}$

Simplify the right side.

Divide $64$ by $16$ .

$r = 4$

Solve for r square root of r+12=2+ square root of r

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