Simplify (-3(m^2-n^2))/( square root of n^2+m^2)+ square root of n^2+m^2

$\frac{- 3 (m^{2} - n^{2})}{\sqrt{n^{2} + m^{2}}} + \sqrt{n^{2} + m^{2}}$

Step 1

Simplify each term.

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

$\frac{- 3 (m + n) (m - n)}{\sqrt{n^{2} + m^{2}}} + \sqrt{n^{2} + m^{2}}$

Move the negative in front of the fraction.

$- \frac{((3) (m + n)) (m - n)}{\sqrt{n^{2} + m^{2}}} + \sqrt{n^{2} + m^{2}}$

Multiply $\frac{3 (m + n) (m - n)}{\sqrt{n^{2} + m^{2}}}$ by $\frac{\sqrt{n^{2} + m^{2}}}{\sqrt{n^{2} + m^{2}}}$ .

$- (\frac{3 (m + n) (m - n)}{\sqrt{n^{2} + m^{2}}} \cdot \frac{\sqrt{n^{2} + m^{2}}}{\sqrt{n^{2} + m^{2}}}) + \sqrt{n^{2} + m^{2}}$

Combine and simplify the denominator.

Multiply $\frac{3 (m + n) (m - n)}{\sqrt{n^{2} + m^{2}}}$ by $\frac{\sqrt{n^{2} + m^{2}}}{\sqrt{n^{2} + m^{2}}}$ .

$- \frac{3 (m + n) (m - n) \sqrt{n^{2} + m^{2}}}{\sqrt{n^{2} + m^{2}} \sqrt{n^{2} + m^{2}}} + \sqrt{n^{2} + m^{2}}$

Raise $\sqrt{n^{2} + m^{2}}$ to the power of $1$ .

$- \frac{3 (m + n) (m - n) \sqrt{n^{2} + m^{2}}}{{\sqrt{n^{2} + m^{2}}}^{1} \sqrt{n^{2} + m^{2}}} + \sqrt{n^{2} + m^{2}}$

Raise $\sqrt{n^{2} + m^{2}}$ to the power of $1$ .

$- \frac{3 (m + n) (m - n) \sqrt{n^{2} + m^{2}}}{{\sqrt{n^{2} + m^{2}}}^{1} {\sqrt{n^{2} + m^{2}}}^{1}} + \sqrt{n^{2} + m^{2}}$

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

$- \frac{3 (m + n) (m - n) \sqrt{n^{2} + m^{2}}}{{\sqrt{n^{2} + m^{2}}}^{1 + 1}} + \sqrt{n^{2} + m^{2}}$

Add $1$ and $1$ .

$- \frac{3 (m + n) (m - n) \sqrt{n^{2} + m^{2}}}{{\sqrt{n^{2} + m^{2}}}^{2}} + \sqrt{n^{2} + m^{2}}$

Rewrite ${\sqrt{n^{2} + m^{2}}}^{2}$ as $n^{2} + m^{2}$ .

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

$- \frac{3 (m + n) (m - n) \sqrt{n^{2} + m^{2}}}{{({(n^{2} + m^{2})}^{\frac{1}{2}})}^{2}} + \sqrt{n^{2} + m^{2}}$

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

$- \frac{3 (m + n) (m - n) \sqrt{n^{2} + m^{2}}}{{(n^{2} + m^{2})}^{\frac{1}{2} \cdot 2}} + \sqrt{n^{2} + m^{2}}$

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

$- \frac{3 (m + n) (m - n) \sqrt{n^{2} + m^{2}}}{{(n^{2} + m^{2})}^{\frac{2}{2}}} + \sqrt{n^{2} + m^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$- \frac{3 (m + n) (m - n) \sqrt{n^{2} + m^{2}}}{{(n^{2} + m^{2})}^{\frac{2}{2}}} + \sqrt{n^{2} + m^{2}}$

Rewrite the expression.

$- \frac{3 (m + n) (m - n) \sqrt{n^{2} + m^{2}}}{{(n^{2} + m^{2})}^{1}} + \sqrt{n^{2} + m^{2}}$

Simplify.

$- \frac{3 (m + n) (m - n) \sqrt{n^{2} + m^{2}}}{n^{2} + m^{2}} + \sqrt{n^{2} + m^{2}}$

Step 2

To write $\sqrt{n^{2} + m^{2}}$ as a fraction with a common denominator, multiply by $\frac{n^{2} + m^{2}}{n^{2} + m^{2}}$ .

$- \frac{3 (m + n) (m - n) \sqrt{n^{2} + m^{2}}}{n^{2} + m^{2}} + \frac{\sqrt{n^{2} + m^{2}} (n^{2} + m^{2})}{n^{2} + m^{2}}$

Step 3

Combine the numerators over the common denominator.

$\frac{- 3 (m + n) (m - n) \sqrt{n^{2} + m^{2}} + \sqrt{n^{2} + m^{2}} (n^{2} + m^{2})}{n^{2} + m^{2}}$

Step 4

Simplify the numerator.

Factor $\sqrt{n^{2} + m^{2}}$ out of $- 3 (m + n) (m - n) \sqrt{n^{2} + m^{2}} + \sqrt{n^{2} + m^{2}} (n^{2} + m^{2})$ .

Factor $\sqrt{n^{2} + m^{2}}$ out of $- 3 (m + n) (m - n) \sqrt{n^{2} + m^{2}}$ .

$\frac{\sqrt{n^{2} + m^{2}} (- 3 (m + n) (m - n)) + \sqrt{n^{2} + m^{2}} (n^{2} + m^{2})}{n^{2} + m^{2}}$

Factor $\sqrt{n^{2} + m^{2}}$ out of $\sqrt{n^{2} + m^{2}} (- 3 (m + n) (m - n)) + \sqrt{n^{2} + m^{2}} (n^{2} + m^{2})$ .

$\frac{\sqrt{n^{2} + m^{2}} (- 3 (m + n) (m - n) + n^{2} + m^{2})}{n^{2} + m^{2}}$

Apply the distributive property.

$\frac{\sqrt{n^{2} + m^{2}} ((- 3 m - 3 n) (m - n) + n^{2} + m^{2})}{n^{2} + m^{2}}$

Expand $(- 3 m - 3 n) (m - n)$ using the FOIL Method.

Apply the distributive property.

$\frac{\sqrt{n^{2} + m^{2}} (- 3 m (m - n) - 3 n (m - n) + n^{2} + m^{2})}{n^{2} + m^{2}}$

Apply the distributive property.

$\frac{\sqrt{n^{2} + m^{2}} (- 3 m \cdot m - 3 m (- n) - 3 n (m - n) + n^{2} + m^{2})}{n^{2} + m^{2}}$

Apply the distributive property.

$\frac{\sqrt{n^{2} + m^{2}} (- 3 m \cdot m - 3 m (- n) - 3 n m - 3 n (- n) + n^{2} + m^{2})}{n^{2} + m^{2}}$

Simplify and combine like terms.

Simplify each term.

Multiply $m$ by $m$ by adding the exponents.

Move $m$ .

$\frac{\sqrt{n^{2} + m^{2}} (- 3 (m \cdot m) - 3 m (- n) - 3 n m - 3 n (- n) + n^{2} + m^{2})}{n^{2} + m^{2}}$

Multiply $m$ by $m$ .

$\frac{\sqrt{n^{2} + m^{2}} (- 3 m^{2} - 3 m (- n) - 3 n m - 3 n (- n) + n^{2} + m^{2})}{n^{2} + m^{2}}$

Rewrite using the commutative property of multiplication.

$\frac{\sqrt{n^{2} + m^{2}} (- 3 m^{2} - 3 \cdot - 1 m n - 3 n m - 3 n (- n) + n^{2} + m^{2})}{n^{2} + m^{2}}$

Multiply $- 3$ by $- 1$ .

$\frac{\sqrt{n^{2} + m^{2}} (- 3 m^{2} + 3 m n - 3 n m - 3 n (- n) + n^{2} + m^{2})}{n^{2} + m^{2}}$

Rewrite using the commutative property of multiplication.

$\frac{\sqrt{n^{2} + m^{2}} (- 3 m^{2} + 3 m n - 3 n m - 3 \cdot - 1 n \cdot n + n^{2} + m^{2})}{n^{2} + m^{2}}$

Multiply $n$ by $n$ by adding the exponents.

Move $n$ .

$\frac{\sqrt{n^{2} + m^{2}} (- 3 m^{2} + 3 m n - 3 n m - 3 \cdot - 1 (n \cdot n) + n^{2} + m^{2})}{n^{2} + m^{2}}$

Multiply $n$ by $n$ .

$\frac{\sqrt{n^{2} + m^{2}} (- 3 m^{2} + 3 m n - 3 n m - 3 \cdot - 1 n^{2} + n^{2} + m^{2})}{n^{2} + m^{2}}$

Multiply $- 3$ by $- 1$ .

$\frac{\sqrt{n^{2} + m^{2}} (- 3 m^{2} + 3 m n - 3 n m + 3 n^{2} + n^{2} + m^{2})}{n^{2} + m^{2}}$

Subtract $3 n m$ from $3 m n$ .

Move $n$ .

$\frac{\sqrt{n^{2} + m^{2}} (- 3 m^{2} + 3 m n - 3 m n + 3 n^{2} + n^{2} + m^{2})}{n^{2} + m^{2}}$

Subtract $3 m n$ from $3 m n$ .

$\frac{\sqrt{n^{2} + m^{2}} (- 3 m^{2} + 0 + 3 n^{2} + n^{2} + m^{2})}{n^{2} + m^{2}}$

Add $- 3 m^{2}$ and $0$ .

$\frac{\sqrt{n^{2} + m^{2}} (- 3 m^{2} + 3 n^{2} + n^{2} + m^{2})}{n^{2} + m^{2}}$

Add $- 3 m^{2}$ and $m^{2}$ .

$\frac{\sqrt{n^{2} + m^{2}} (- 2 m^{2} + 3 n^{2} + n^{2})}{n^{2} + m^{2}}$

Add $3 n^{2}$ and $n^{2}$ .

$\frac{\sqrt{n^{2} + m^{2}} (- 2 m^{2} + 4 n^{2})}{n^{2} + m^{2}}$

Factor $2$ out of $- 2 m^{2} + 4 n^{2}$ .

Factor $2$ out of $- 2 m^{2}$ .

$\frac{\sqrt{n^{2} + m^{2}} (2 (- m^{2}) + 4 n^{2})}{n^{2} + m^{2}}$

Factor $2$ out of $4 n^{2}$ .

$\frac{\sqrt{n^{2} + m^{2}} (2 (- m^{2}) + 2 (2 n^{2}))}{n^{2} + m^{2}}$

Factor $2$ out of $2 (- m^{2}) + 2 (2 n^{2})$ .

$\frac{\sqrt{n^{2} + m^{2}} (2 (- m^{2} + 2 n^{2}))}{n^{2} + m^{2}}$

$\frac{\sqrt{n^{2} + m^{2}} \cdot 2 (- m^{2} + 2 n^{2})}{n^{2} + m^{2}}$

Step 5

Simplify with factoring out.

Move $2$ to the left of $\sqrt{n^{2} + m^{2}}$ .

$\frac{2 \cdot \sqrt{n^{2} + m^{2}} (- m^{2} + 2 n^{2})}{n^{2} + m^{2}}$

Factor $- 1$ out of $- m^{2}$ .

$\frac{2 \sqrt{n^{2} + m^{2}} (- (m^{2}) + 2 n^{2})}{n^{2} + m^{2}}$

Factor $- 1$ out of $2 n^{2}$ .

$\frac{2 \sqrt{n^{2} + m^{2}} (- (m^{2}) - (- 2 n^{2}))}{n^{2} + m^{2}}$

Factor $- 1$ out of $- (m^{2}) - (- 2 n^{2})$ .

$\frac{2 \sqrt{n^{2} + m^{2}} (- (m^{2} - 2 n^{2}))}{n^{2} + m^{2}}$

Simplify the expression.

Rewrite $- (m^{2} - 2 n^{2})$ as $- 1 (m^{2} - 2 n^{2})$ .

$\frac{2 \sqrt{n^{2} + m^{2}} (- 1 (m^{2} - 2 n^{2}))}{n^{2} + m^{2}}$

Move the negative in front of the fraction.

$- \frac{(2 \sqrt{n^{2} + m^{2}}) (m^{2} - 2 n^{2})}{n^{2} + m^{2}}$

Reorder factors in $- \frac{(2 \sqrt{n^{2} + m^{2}}) (m^{2} - 2 n^{2})}{n^{2} + m^{2}}$ .

$- \frac{2 \sqrt{n^{2} + m^{2}} (m^{2} - 2 n^{2})}{n^{2} + m^{2}}$

Simplify (-3(m^2-n^2))/( square root of n^2+m^2)+ square root of n^2+m^2

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