Basic Math Examples

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

Step 1

Combine fractions.

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Step 1.1

Combine $b^{2}$ and $\frac{1}{4}$ .

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

Step 1.2

Rewrite ${(4 a^{2} - \frac{b^{2}}{4})}^{2}$ as $(4 a^{2} - \frac{b^{2}}{4}) (4 a^{2} - \frac{b^{2}}{4})$ .

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

Step 2

Expand $(4 a^{2} - \frac{b^{2}}{4}) (4 a^{2} - \frac{b^{2}}{4})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.1.2

Multiply $a^{2}$ by $a^{2}$ by adding the exponents.

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Step 3.1.2.1

Move $a^{2}$ .

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

Step 3.1.2.2

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

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

Step 3.1.2.3

Add $2$ and $2$ .

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

Step 3.1.3

Multiply $4$ by $4$ .

$16 a^{4} + 4 a^{2} (- \frac{b^{2}}{4}) - \frac{b^{2}}{4} (4 a^{2}) - \frac{b^{2}}{4} (- \frac{b^{2}}{4})$

Step 3.1.4

Cancel the common factor of $4$ .

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Step 3.1.4.1

Move the leading negative in $- \frac{b^{2}}{4}$ into the numerator.

$16 a^{4} + 4 a^{2} \frac{- b^{2}}{4} - \frac{b^{2}}{4} (4 a^{2}) - \frac{b^{2}}{4} (- \frac{b^{2}}{4})$

Step 3.1.4.2

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

$16 a^{4} + 4 (a^{2}) \frac{- b^{2}}{4} - \frac{b^{2}}{4} (4 a^{2}) - \frac{b^{2}}{4} (- \frac{b^{2}}{4})$

Step 3.1.4.3

Cancel the common factor.

$16 a^{4} + 4 a^{2} \frac{- b^{2}}{4} - \frac{b^{2}}{4} (4 a^{2}) - \frac{b^{2}}{4} (- \frac{b^{2}}{4})$

Step 3.1.4.4

Rewrite the expression.

$16 a^{4} + a^{2} (- b^{2}) - \frac{b^{2}}{4} (4 a^{2}) - \frac{b^{2}}{4} (- \frac{b^{2}}{4})$

Step 3.1.5

Rewrite using the commutative property of multiplication.

$16 a^{4} - a^{2} b^{2} - \frac{b^{2}}{4} (4 a^{2}) - \frac{b^{2}}{4} (- \frac{b^{2}}{4})$

Step 3.1.6

Cancel the common factor of $4$ .

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Step 3.1.6.1

Move the leading negative in $- \frac{b^{2}}{4}$ into the numerator.

$16 a^{4} - a^{2} b^{2} + \frac{- b^{2}}{4} (4 a^{2}) - \frac{b^{2}}{4} (- \frac{b^{2}}{4})$

Step 3.1.6.2

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

$16 a^{4} - a^{2} b^{2} + \frac{- b^{2}}{4} (4 (a^{2})) - \frac{b^{2}}{4} (- \frac{b^{2}}{4})$

Step 3.1.6.3

Cancel the common factor.

$16 a^{4} - a^{2} b^{2} + \frac{- b^{2}}{4} (4 a^{2}) - \frac{b^{2}}{4} (- \frac{b^{2}}{4})$

Step 3.1.6.4

Rewrite the expression.

$16 a^{4} - a^{2} b^{2} - b^{2} a^{2} - \frac{b^{2}}{4} (- \frac{b^{2}}{4})$

Step 3.1.7

Multiply $- \frac{b^{2}}{4} (- \frac{b^{2}}{4})$ .

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Step 3.1.7.1

Multiply $- 1$ by $- 1$ .

$16 a^{4} - a^{2} b^{2} - b^{2} a^{2} + 1 \frac{b^{2}}{4} \frac{b^{2}}{4}$

Step 3.1.7.2

Multiply $\frac{b^{2}}{4}$ by $1$ .

$16 a^{4} - a^{2} b^{2} - b^{2} a^{2} + \frac{b^{2}}{4} \cdot \frac{b^{2}}{4}$

Step 3.1.7.3

Multiply $\frac{b^{2}}{4}$ by $\frac{b^{2}}{4}$ .

$16 a^{4} - a^{2} b^{2} - b^{2} a^{2} + \frac{b^{2} b^{2}}{4 \cdot 4}$

Step 3.1.7.4

Multiply $b^{2}$ by $b^{2}$ by adding the exponents.

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Step 3.1.7.4.1

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

$16 a^{4} - a^{2} b^{2} - b^{2} a^{2} + \frac{b^{2 + 2}}{4 \cdot 4}$

Step 3.1.7.4.2

Add $2$ and $2$ .

$16 a^{4} - a^{2} b^{2} - b^{2} a^{2} + \frac{b^{4}}{4 \cdot 4}$

Step 3.1.7.5

Multiply $4$ by $4$ .

$16 a^{4} - a^{2} b^{2} - b^{2} a^{2} + \frac{b^{4}}{16}$

Step 3.2

Subtract $b^{2} a^{2}$ from $- a^{2} b^{2}$ .

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Step 3.2.1

Move $b^{2}$ .

$16 a^{4} - a^{2} b^{2} - 1 a^{2} b^{2} + \frac{b^{4}}{16}$

Step 3.2.2

Subtract $a^{2} b^{2}$ from $- a^{2} b^{2}$ .

$16 a^{4} - 2 a^{2} b^{2} + \frac{b^{4}}{16}$