Simplify ( square root of m- square root of n)^2

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

Step 1

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

$(\sqrt{m} - \sqrt{n}) (\sqrt{m} - \sqrt{n})$

Step 2

Expand $(\sqrt{m} - \sqrt{n}) (\sqrt{m} - \sqrt{n})$ using the FOIL Method.

Apply the distributive property.

$\sqrt{m} (\sqrt{m} - \sqrt{n}) - \sqrt{n} (\sqrt{m} - \sqrt{n})$

Apply the distributive property.

$\sqrt{m} \sqrt{m} + \sqrt{m} (- \sqrt{n}) - \sqrt{n} (\sqrt{m} - \sqrt{n})$

Apply the distributive property.

$\sqrt{m} \sqrt{m} + \sqrt{m} (- \sqrt{n}) - \sqrt{n} \sqrt{m} - \sqrt{n} (- \sqrt{n})$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{m} \sqrt{m}$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$m + \sqrt{m} (- \sqrt{n}) - \sqrt{n} \sqrt{m} - \sqrt{n} (- \sqrt{n})$

Rewrite using the commutative property of multiplication.

$m - \sqrt{m} \sqrt{n} - \sqrt{n} \sqrt{m} - \sqrt{n} (- \sqrt{n})$

Combine using the product rule for radicals.

$m - \sqrt{n m} - \sqrt{n} \sqrt{m} - \sqrt{n} (- \sqrt{n})$

Combine using the product rule for radicals.

$m - \sqrt{n m} - \sqrt{m n} - \sqrt{n} (- \sqrt{n})$

Multiply $- \sqrt{n} (- \sqrt{n})$ .

Multiply $- 1$ by $- 1$ .

$m - \sqrt{n m} - \sqrt{m n} + 1 \sqrt{n} \sqrt{n}$

Multiply $\sqrt{n}$ by $1$ .

$m - \sqrt{n m} - \sqrt{m n} + \sqrt{n} \sqrt{n}$

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

$m - \sqrt{n m} - \sqrt{m n} + {\sqrt{n}}^{1} \sqrt{n}$

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

$m - \sqrt{n m} - \sqrt{m n} + {\sqrt{n}}^{1} {\sqrt{n}}^{1}$

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

$m - \sqrt{n m} - \sqrt{m n} + {\sqrt{n}}^{1 + 1}$

Add $1$ and $1$ .

$m - \sqrt{n m} - \sqrt{m n} + {\sqrt{n}}^{2}$

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$m - \sqrt{n m} - \sqrt{m n} + n^{1}$

Simplify.

$m - \sqrt{n m} - \sqrt{m n} + n$

Subtract $\sqrt{m n}$ from $- \sqrt{n m}$ .

Reorder $n$ and $m$ .

$m - \sqrt{m n} - \sqrt{m n} + n$

Subtract $\sqrt{m n}$ from $- \sqrt{m n}$ .

$m - 2 \sqrt{m n} + n$

Step 4

Move $- 2 \sqrt{m n}$ .

$m + n - 2 \sqrt{m n}$

Simplify ( square root of m- square root of n)^2

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