Simplify ( square root of a- square root of b)^2

${(\sqrt{a} - \sqrt{b})}^{2}$

Step 1

Rewrite ${(\sqrt{a} - \sqrt{b})}^{2}$ as $(\sqrt{a} - \sqrt{b}) (\sqrt{a} - \sqrt{b})$ .

$(\sqrt{a} - \sqrt{b}) (\sqrt{a} - \sqrt{b})$

Step 2

Expand $(\sqrt{a} - \sqrt{b}) (\sqrt{a} - \sqrt{b})$ using the FOIL Method.

Apply the distributive property.

$\sqrt{a} (\sqrt{a} - \sqrt{b}) - \sqrt{b} (\sqrt{a} - \sqrt{b})$

Apply the distributive property.

$\sqrt{a} \sqrt{a} + \sqrt{a} (- \sqrt{b}) - \sqrt{b} (\sqrt{a} - \sqrt{b})$

Apply the distributive property.

$\sqrt{a} \sqrt{a} + \sqrt{a} (- \sqrt{b}) - \sqrt{b} \sqrt{a} - \sqrt{b} (- \sqrt{b})$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{a} \sqrt{a}$ .

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

${\sqrt{a}}^{1} \sqrt{a} + \sqrt{a} (- \sqrt{b}) - \sqrt{b} \sqrt{a} - \sqrt{b} (- \sqrt{b})$

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

${\sqrt{a}}^{1} {\sqrt{a}}^{1} + \sqrt{a} (- \sqrt{b}) - \sqrt{b} \sqrt{a} - \sqrt{b} (- \sqrt{b})$

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

${\sqrt{a}}^{1 + 1} + \sqrt{a} (- \sqrt{b}) - \sqrt{b} \sqrt{a} - \sqrt{b} (- \sqrt{b})$

Add $1$ and $1$ .

${\sqrt{a}}^{2} + \sqrt{a} (- \sqrt{b}) - \sqrt{b} \sqrt{a} - \sqrt{b} (- \sqrt{b})$

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

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

${(a^{\frac{1}{2}})}^{2} + \sqrt{a} (- \sqrt{b}) - \sqrt{b} \sqrt{a} - \sqrt{b} (- \sqrt{b})$

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

$a^{\frac{1}{2} \cdot 2} + \sqrt{a} (- \sqrt{b}) - \sqrt{b} \sqrt{a} - \sqrt{b} (- \sqrt{b})$

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

$a^{\frac{2}{2}} + \sqrt{a} (- \sqrt{b}) - \sqrt{b} \sqrt{a} - \sqrt{b} (- \sqrt{b})$

Cancel the common factor of $2$ .

Cancel the common factor.

$a^{\frac{2}{2}} + \sqrt{a} (- \sqrt{b}) - \sqrt{b} \sqrt{a} - \sqrt{b} (- \sqrt{b})$

Rewrite the expression.

$a^{1} + \sqrt{a} (- \sqrt{b}) - \sqrt{b} \sqrt{a} - \sqrt{b} (- \sqrt{b})$

Simplify.

$a + \sqrt{a} (- \sqrt{b}) - \sqrt{b} \sqrt{a} - \sqrt{b} (- \sqrt{b})$

Rewrite using the commutative property of multiplication.

$a - \sqrt{a} \sqrt{b} - \sqrt{b} \sqrt{a} - \sqrt{b} (- \sqrt{b})$

Combine using the product rule for radicals.

$a - \sqrt{b a} - \sqrt{b} \sqrt{a} - \sqrt{b} (- \sqrt{b})$

Combine using the product rule for radicals.

$a - \sqrt{b a} - \sqrt{a b} - \sqrt{b} (- \sqrt{b})$

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

Multiply $- 1$ by $- 1$ .

$a - \sqrt{b a} - \sqrt{a b} + 1 \sqrt{b} \sqrt{b}$

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

$a - \sqrt{b a} - \sqrt{a b} + \sqrt{b} \sqrt{b}$

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

$a - \sqrt{b a} - \sqrt{a b} + {\sqrt{b}}^{1} \sqrt{b}$

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

$a - \sqrt{b a} - \sqrt{a b} + {\sqrt{b}}^{1} {\sqrt{b}}^{1}$

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

$a - \sqrt{b a} - \sqrt{a b} + {\sqrt{b}}^{1 + 1}$

Add $1$ and $1$ .

$a - \sqrt{b a} - \sqrt{a b} + {\sqrt{b}}^{2}$

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

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

$a - \sqrt{b a} - \sqrt{a b} + {(b^{\frac{1}{2}})}^{2}$

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

$a - \sqrt{b a} - \sqrt{a b} + b^{\frac{1}{2} \cdot 2}$

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

$a - \sqrt{b a} - \sqrt{a b} + b^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$a - \sqrt{b a} - \sqrt{a b} + b^{\frac{2}{2}}$

Rewrite the expression.

$a - \sqrt{b a} - \sqrt{a b} + b^{1}$

Simplify.

$a - \sqrt{b a} - \sqrt{a b} + b$

Subtract $\sqrt{a b}$ from $- \sqrt{b a}$ .

Reorder $b$ and $a$ .

$a - \sqrt{a b} - \sqrt{a b} + b$

Subtract $\sqrt{a b}$ from $- \sqrt{a b}$ .

$a - 2 \sqrt{a b} + b$

Step 4

Move $- 2 \sqrt{a b}$ .

$a + b - 2 \sqrt{a b}$

Simplify ( square root of a- square root of b)^2

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