Pre-Algebra Examples

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

Step 1

Expand $(2 \sqrt{a} + \sqrt{b}) (2 \sqrt{a} - \sqrt{b})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $2 \sqrt{a} (2 \sqrt{a})$ .

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Step 2.1.1.1

Multiply $2$ by $2$ .

$4 \sqrt{a} \sqrt{a} + 2 \sqrt{a} (- \sqrt{b}) + \sqrt{b} (2 \sqrt{a}) + \sqrt{b} (- \sqrt{b})$

Step 2.1.1.2

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

$4 ({\sqrt{a}}^{1} \sqrt{a}) + 2 \sqrt{a} (- \sqrt{b}) + \sqrt{b} (2 \sqrt{a}) + \sqrt{b} (- \sqrt{b})$

Step 2.1.1.3

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

$4 ({\sqrt{a}}^{1} {\sqrt{a}}^{1}) + 2 \sqrt{a} (- \sqrt{b}) + \sqrt{b} (2 \sqrt{a}) + \sqrt{b} (- \sqrt{b})$

Step 2.1.1.4

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

$4 {\sqrt{a}}^{1 + 1} + 2 \sqrt{a} (- \sqrt{b}) + \sqrt{b} (2 \sqrt{a}) + \sqrt{b} (- \sqrt{b})$

Step 2.1.1.5

Add $1$ and $1$ .

$4 {\sqrt{a}}^{2} + 2 \sqrt{a} (- \sqrt{b}) + \sqrt{b} (2 \sqrt{a}) + \sqrt{b} (- \sqrt{b})$

Step 2.1.2

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

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Step 2.1.2.1

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

$4 {(a^{\frac{1}{2}})}^{2} + 2 \sqrt{a} (- \sqrt{b}) + \sqrt{b} (2 \sqrt{a}) + \sqrt{b} (- \sqrt{b})$

Step 2.1.2.2

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

$4 a^{\frac{1}{2} \cdot 2} + 2 \sqrt{a} (- \sqrt{b}) + \sqrt{b} (2 \sqrt{a}) + \sqrt{b} (- \sqrt{b})$

Step 2.1.2.3

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

$4 a^{\frac{2}{2}} + 2 \sqrt{a} (- \sqrt{b}) + \sqrt{b} (2 \sqrt{a}) + \sqrt{b} (- \sqrt{b})$

Step 2.1.2.4

Cancel the common factor of $2$ .

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Step 2.1.2.4.1

Cancel the common factor.

$4 a^{\frac{2}{2}} + 2 \sqrt{a} (- \sqrt{b}) + \sqrt{b} (2 \sqrt{a}) + \sqrt{b} (- \sqrt{b})$

Step 2.1.2.4.2

Rewrite the expression.

$4 a^{1} + 2 \sqrt{a} (- \sqrt{b}) + \sqrt{b} (2 \sqrt{a}) + \sqrt{b} (- \sqrt{b})$

Step 2.1.2.5

Simplify.

$4 a + 2 \sqrt{a} (- \sqrt{b}) + \sqrt{b} (2 \sqrt{a}) + \sqrt{b} (- \sqrt{b})$

Step 2.1.3

Multiply $2 \sqrt{a} (- \sqrt{b})$ .

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Step 2.1.3.1

Multiply $- 1$ by $2$ .

$4 a - 2 \sqrt{a} \sqrt{b} + \sqrt{b} (2 \sqrt{a}) + \sqrt{b} (- \sqrt{b})$

Step 2.1.3.2

Combine using the product rule for radicals.

$4 a - 2 \sqrt{b a} + \sqrt{b} (2 \sqrt{a}) + \sqrt{b} (- \sqrt{b})$

Step 2.1.4

Rewrite using the commutative property of multiplication.

$4 a - 2 \sqrt{b a} + 2 \sqrt{b} \sqrt{a} + \sqrt{b} (- \sqrt{b})$

Step 2.1.5

Combine using the product rule for radicals.

$4 a - 2 \sqrt{b a} + 2 \sqrt{a b} + \sqrt{b} (- \sqrt{b})$

Step 2.1.6

Rewrite using the commutative property of multiplication.

$4 a - 2 \sqrt{b a} + 2 \sqrt{a b} - \sqrt{b} \sqrt{b}$

Step 2.1.7

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

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Step 2.1.7.1

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

$4 a - 2 \sqrt{b a} + 2 \sqrt{a b} - ({\sqrt{b}}^{1} \sqrt{b})$

Step 2.1.7.2

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

$4 a - 2 \sqrt{b a} + 2 \sqrt{a b} - ({\sqrt{b}}^{1} {\sqrt{b}}^{1})$

Step 2.1.7.3

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

$4 a - 2 \sqrt{b a} + 2 \sqrt{a b} - {\sqrt{b}}^{1 + 1}$

Step 2.1.7.4

Add $1$ and $1$ .

$4 a - 2 \sqrt{b a} + 2 \sqrt{a b} - {\sqrt{b}}^{2}$

Step 2.1.8

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

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Step 2.1.8.1

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

$4 a - 2 \sqrt{b a} + 2 \sqrt{a b} - {(b^{\frac{1}{2}})}^{2}$

Step 2.1.8.2

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

$4 a - 2 \sqrt{b a} + 2 \sqrt{a b} - b^{\frac{1}{2} \cdot 2}$

Step 2.1.8.3

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

$4 a - 2 \sqrt{b a} + 2 \sqrt{a b} - b^{\frac{2}{2}}$

Step 2.1.8.4

Cancel the common factor of $2$ .

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Step 2.1.8.4.1

Cancel the common factor.

$4 a - 2 \sqrt{b a} + 2 \sqrt{a b} - b^{\frac{2}{2}}$

Step 2.1.8.4.2

Rewrite the expression.

$4 a - 2 \sqrt{b a} + 2 \sqrt{a b} - b^{1}$

Step 2.1.8.5

Simplify.

$4 a - 2 \sqrt{b a} + 2 \sqrt{a b} - b$

Step 2.2

Add $- 2 \sqrt{b a}$ and $2 \sqrt{a b}$ .

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Step 2.2.1

Reorder $b$ and $a$ .

$4 a - 2 \sqrt{a b} + 2 \sqrt{a b} - b$

Step 2.2.2

Add $- 2 \sqrt{a b}$ and $2 \sqrt{a b}$ .

$4 a + 0 - b$

Step 2.3

Add $4 a$ and $0$ .

$4 a - b$