Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x + y$ and $b = x - y$ .

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

Step 1.2

Simplify.

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Step 1.2.1

Add $x$ and $x$ .

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

Step 1.2.2

Subtract $y$ from $y$ .

$M = {(\frac{(2 x + 0) (x + y - (x - y))}{x y})}^{\frac{1}{2}}$

Step 1.2.3

Add $2 x$ and $0$ .

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

Step 1.2.4

Apply the distributive property.

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

Step 1.2.5

Multiply $- - y$ .

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Step 1.2.5.1

Multiply $- 1$ by $- 1$ .

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

Step 1.2.5.2

Multiply $y$ by $1$ .

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

Step 1.2.6

Subtract $x$ from $x$ .

$M = {(\frac{2 x (y + 0 + y)}{x y})}^{\frac{1}{2}}$

Step 1.2.7

Add $y$ and $0$ .

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

Step 1.2.8

Add $y$ and $y$ .

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

Step 1.2.9

Multiply $2$ by $2$ .

$M = {(\frac{4 x y}{x y})}^{\frac{1}{2}}$

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of $x$ .

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Step 2.1.1

Cancel the common factor.

$M = {(\frac{4 x y}{x y})}^{\frac{1}{2}}$

Step 2.1.2

Rewrite the expression.

$M = {(\frac{4 y}{y})}^{\frac{1}{2}}$

Step 2.2

Cancel the common factor of $y$ .

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Step 2.2.1

Cancel the common factor.

$M = {(\frac{4 y}{y})}^{\frac{1}{2}}$

Step 2.2.2

Divide $4$ by $1$ .

$M = 4^{\frac{1}{2}}$

Step 2.3

Simplify the expression.

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Step 2.3.1

Rewrite $4$ as $2^{2}$ .

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

Step 2.3.2

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

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

Step 2.4

Cancel the common factor of $2$ .

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Step 2.4.1

Cancel the common factor.

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

Step 2.4.2

Rewrite the expression.

$M = 2^{1}$

Step 3

Evaluate the exponent.

$M = 2$