Solve for u square root of u+1 = square root of u+21

$\sqrt{u} + 1 = \sqrt{u + 21}$

Subtract $1$ from both sides of the equation.

$\sqrt{u} = \sqrt{u + 21} - 1$

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

${\sqrt{u}}^{2} = {(\sqrt{u + 21} - 1)}^{2}$

Simplify each side of the equation.

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

${(u^{\frac{1}{2}})}^{2} = {(\sqrt{u + 21} - 1)}^{2}$

Simplify the left side.

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

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

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

$u^{\frac{1}{2} \cdot 2} = {(\sqrt{u + 21} - 1)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$u^{\frac{1}{2} \cdot 2} = {(\sqrt{u + 21} - 1)}^{2}$

Rewrite the expression.

$u^{1} = {(\sqrt{u + 21} - 1)}^{2}$

Simplify.

$u = {(\sqrt{u + 21} - 1)}^{2}$

Simplify the right side.

Simplify ${(\sqrt{u + 21} - 1)}^{2}$ .

Rewrite ${(\sqrt{u + 21} - 1)}^{2}$ as $(\sqrt{u + 21} - 1) (\sqrt{u + 21} - 1)$ .

$u = (\sqrt{u + 21} - 1) (\sqrt{u + 21} - 1)$

Expand $(\sqrt{u + 21} - 1) (\sqrt{u + 21} - 1)$ using the FOIL Method.

Apply the distributive property.

$u = \sqrt{u + 21} (\sqrt{u + 21} - 1) - 1 (\sqrt{u + 21} - 1)$

Apply the distributive property.

$u = \sqrt{u + 21} \sqrt{u + 21} + \sqrt{u + 21} \cdot - 1 - 1 (\sqrt{u + 21} - 1)$

Apply the distributive property.

$u = \sqrt{u + 21} \sqrt{u + 21} + \sqrt{u + 21} \cdot - 1 - 1 \sqrt{u + 21} - 1 \cdot - 1$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{u + 21} \sqrt{u + 21}$ .

Raise $\sqrt{u + 21}$ to the power of $1$ .

$u = {\sqrt{u + 21}}^{1} \sqrt{u + 21} + \sqrt{u + 21} \cdot - 1 - 1 \sqrt{u + 21} - 1 \cdot - 1$

Raise $\sqrt{u + 21}$ to the power of $1$ .

$u = {\sqrt{u + 21}}^{1} {\sqrt{u + 21}}^{1} + \sqrt{u + 21} \cdot - 1 - 1 \sqrt{u + 21} - 1 \cdot - 1$

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

$u = {\sqrt{u + 21}}^{1 + 1} + \sqrt{u + 21} \cdot - 1 - 1 \sqrt{u + 21} - 1 \cdot - 1$

Add $1$ and $1$ .

$u = {\sqrt{u + 21}}^{2} + \sqrt{u + 21} \cdot - 1 - 1 \sqrt{u + 21} - 1 \cdot - 1$

Rewrite ${\sqrt{u + 21}}^{2}$ as $u + 21$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u + 21}$ as ${(u + 21)}^{\frac{1}{2}}$ .

$u = {({(u + 21)}^{\frac{1}{2}})}^{2} + \sqrt{u + 21} \cdot - 1 - 1 \sqrt{u + 21} - 1 \cdot - 1$

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

$u = {(u + 21)}^{\frac{1}{2} \cdot 2} + \sqrt{u + 21} \cdot - 1 - 1 \sqrt{u + 21} - 1 \cdot - 1$

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

$u = {(u + 21)}^{\frac{2}{2}} + \sqrt{u + 21} \cdot - 1 - 1 \sqrt{u + 21} - 1 \cdot - 1$

Cancel the common factor of $2$ .

Cancel the common factor.

$u = {(u + 21)}^{\frac{2}{2}} + \sqrt{u + 21} \cdot - 1 - 1 \sqrt{u + 21} - 1 \cdot - 1$

Rewrite the expression.

$u = {(u + 21)}^{1} + \sqrt{u + 21} \cdot - 1 - 1 \sqrt{u + 21} - 1 \cdot - 1$

Simplify.

$u = u + 21 + \sqrt{u + 21} \cdot - 1 - 1 \sqrt{u + 21} - 1 \cdot - 1$

Move $- 1$ to the left of $\sqrt{u + 21}$ .

$u = u + 21 - 1 \cdot \sqrt{u + 21} - 1 \sqrt{u + 21} - 1 \cdot - 1$

Rewrite $- 1 \sqrt{u + 21}$ as $- \sqrt{u + 21}$ .

$u = u + 21 - \sqrt{u + 21} - 1 \sqrt{u + 21} - 1 \cdot - 1$

Rewrite $- 1 \sqrt{u + 21}$ as $- \sqrt{u + 21}$ .

$u = u + 21 - \sqrt{u + 21} - \sqrt{u + 21} - 1 \cdot - 1$

Multiply $- 1$ by $- 1$ .

$u = u + 21 - \sqrt{u + 21} - \sqrt{u + 21} + 1$

Add $21$ and $1$ .

$u = u + 22 - \sqrt{u + 21} - \sqrt{u + 21}$

Subtract $\sqrt{u + 21}$ from $- \sqrt{u + 21}$ .

$u = u + 22 - 2 \sqrt{u + 21}$

Solve for $- 2 \sqrt{u + 21}$ .

Rewrite the equation as $u + 22 - 2 \sqrt{u + 21} = u$ .

$u + 22 - 2 \sqrt{u + 21} = u$

Move all terms not containing $- 2 \sqrt{u + 21}$ to the right side of the equation.

Subtract $u$ from both sides of the equation.

$22 - 2 \sqrt{u + 21} = u - u$

Subtract $22$ from both sides of the equation.

$- 2 \sqrt{u + 21} = u - u - 22$

Subtract $u$ from $u$ .

$- 2 \sqrt{u + 21} = 0 - 22$

Subtract $22$ from $0$ .

$- 2 \sqrt{u + 21} = - 22$

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

${(- 2 \sqrt{u + 21})}^{2} = {(- 22)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u + 21}$ as ${(u + 21)}^{\frac{1}{2}}$ .

${(- 2 {(u + 21)}^{\frac{1}{2}})}^{2} = {(- 22)}^{2}$

Simplify the left side.

Simplify ${(- 2 {(u + 21)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 2 {(u + 21)}^{\frac{1}{2}}$ .

${(- 2)}^{2} {({(u + 21)}^{\frac{1}{2}})}^{2} = {(- 22)}^{2}$

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

$4 {({(u + 21)}^{\frac{1}{2}})}^{2} = {(- 22)}^{2}$

Multiply the exponents in ${({(u + 21)}^{\frac{1}{2}})}^{2}$ .

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

$4 {(u + 21)}^{\frac{1}{2} \cdot 2} = {(- 22)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {(u + 21)}^{\frac{1}{2} \cdot 2} = {(- 22)}^{2}$

Rewrite the expression.

$4 {(u + 21)}^{1} = {(- 22)}^{2}$

Simplify.

$4 (u + 21) = {(- 22)}^{2}$

Apply the distributive property.

$4 u + 4 \cdot 21 = {(- 22)}^{2}$

Multiply $4$ by $21$ .

$4 u + 84 = {(- 22)}^{2}$

Simplify the right side.

Raise $- 22$ to the power of $2$ .

$4 u + 84 = 484$

Solve for $u$ .

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

Subtract $84$ from both sides of the equation.

$4 u = 484 - 84$

Subtract $84$ from $484$ .

$4 u = 400$

Divide each term in $4 u = 400$ by $4$ and simplify.

Divide each term in $4 u = 400$ by $4$ .

$\frac{4 u}{4} = \frac{400}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 u}{4} = \frac{400}{4}$

Divide $u$ by $1$ .

$u = \frac{400}{4}$

Simplify the right side.

Divide $400$ by $4$ .

$u = 100$

Solve for u square root of u+1 = square root of u+21

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