Evaluate square root of x^2+y^2=x+y

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

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$x^{2} + y^{2} = {(x + y)}^{2}$

Simplify the right side.

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

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

$x^{2} + y^{2} = (x + y) (x + y)$

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

Apply the distributive property.

$x^{2} + y^{2} = x (x + y) + y (x + y)$

Apply the distributive property.

$x^{2} + y^{2} = x \cdot x + x y + y (x + y)$

Apply the distributive property.

$x^{2} + y^{2} = x \cdot x + x y + y x + y \cdot y$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + y^{2} = x^{2} + x y + y x + y \cdot y$

Multiply $y$ by $y$ .

$x^{2} + y^{2} = x^{2} + x y + y x + y^{2}$

Add $x y$ and $y x$ .

Reorder $y$ and $x$ .

$x^{2} + y^{2} = x^{2} + x y + x y + y^{2}$

Add $x y$ and $x y$ .

$x^{2} + y^{2} = x^{2} + 2 x y + y^{2}$

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.

$x^{2} + 2 x y + y^{2} = x^{2} + y^{2}$

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

Subtract $x^{2}$ from both sides of the equation.

$x^{2} + 2 x y + y^{2} - x^{2} = y^{2}$

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

Subtract $x^{2}$ from $x^{2}$ .

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

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

$2 x y + y^{2} = y^{2}$

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

Subtract $y^{2}$ from both sides of the equation.

$2 x y = y^{2} - y^{2}$

Subtract $y^{2}$ from $y^{2}$ .

$2 x y = 0$

Divide each term in $2 x y = 0$ by $2 y$ and simplify.

Divide each term in $2 x y = 0$ by $2 y$ .

$\frac{2 x y}{2 y} = \frac{0}{2 y}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x y}{2 y} = \frac{0}{2 y}$

Rewrite the expression.

$\frac{x y}{y} = \frac{0}{2 y}$

Cancel the common factor of $y$ .

Cancel the common factor.

$\frac{x y}{y} = \frac{0}{2 y}$

Divide $x$ by $1$ .

$x = \frac{0}{2 y}$

Simplify the right side.

Cancel the common factor of $0$ and $2$ .

Factor $2$ out of $0$ .

$x = \frac{2 (0)}{2 y}$

Cancel the common factors.

Factor $2$ out of $2 y$ .

$x = \frac{2 (0)}{2 (y)}$

Cancel the common factor.

$x = \frac{2 \cdot 0}{2 y}$

Rewrite the expression.

$x = \frac{0}{y}$

Divide $0$ by $y$ .

$x = 0$

Evaluate square root of x^2+y^2=x+y

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