Pre-Algebra Examples

$\frac{64 a^{2} - 9}{16}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $64 a^{2}$ as ${(8 a)}^{2}$ .

$\frac{{(8 a)}^{2} - 9}{16}$

Step 1.2

Rewrite $9$ as $3^{2}$ .

$\frac{{(8 a)}^{2} - 3^{2}}{16}$

Step 1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 8 a$ and $b = 3$ .

$\frac{(8 a + 3) (8 a - 3)}{16}$

Step 2

Expand $(8 a + 3) (8 a - 3)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{8 a (8 a - 3) + 3 (8 a - 3)}{16}$

Step 2.2

Apply the distributive property.

$\frac{8 a (8 a) + 8 a \cdot - 3 + 3 (8 a - 3)}{16}$

Step 2.3

Apply the distributive property.

$\frac{8 a (8 a) + 8 a \cdot - 3 + 3 (8 a) + 3 \cdot - 3}{16}$

Step 3

Combine the opposite terms in $8 a (8 a) + 8 a \cdot - 3 + 3 (8 a) + 3 \cdot - 3$ .

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Step 3.1

Reorder the factors in the terms $8 a \cdot - 3$ and $3 (8 a)$ .

$\frac{8 a (8 a) - 3 \cdot 8 a + 3 \cdot 8 a + 3 \cdot - 3}{16}$

Step 3.2

Add $- 3 \cdot 8 a$ and $3 \cdot 8 a$ .

$\frac{8 a (8 a) + 0 + 3 \cdot - 3}{16}$

Step 3.3

Add $8 a (8 a)$ and $0$ .

$\frac{8 a (8 a) + 3 \cdot - 3}{16}$

Step 4

Simplify each term.

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Step 4.1

Rewrite using the commutative property of multiplication.

$\frac{8 \cdot 8 a \cdot a + 3 \cdot - 3}{16}$

Step 4.2

Multiply $a$ by $a$ by adding the exponents.

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Step 4.2.1

Move $a$ .

$\frac{8 \cdot 8 (a \cdot a) + 3 \cdot - 3}{16}$

Step 4.2.2

Multiply $a$ by $a$ .

$\frac{8 \cdot 8 a^{2} + 3 \cdot - 3}{16}$

Step 4.3

Multiply $8$ by $8$ .

$\frac{64 a^{2} + 3 \cdot - 3}{16}$

Step 4.4

Multiply $3$ by $- 3$ .

$\frac{64 a^{2} - 9}{16}$

Step 5

Split the fraction $\frac{64 a^{2} - 9}{16}$ into two fractions.

$\frac{64 a^{2}}{16} + \frac{- 9}{16}$

Step 6

Cancel the common factor of $64$ and $16$ .

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Step 6.1

Factor $16$ out of $64 a^{2}$ .

$\frac{16 (4 a^{2})}{16} + \frac{- 9}{16}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $16$ out of $16$ .

$\frac{16 (4 a^{2})}{16 (1)} + \frac{- 9}{16}$

Step 6.2.2

Cancel the common factor.

$\frac{16 (4 a^{2})}{16 \cdot 1} + \frac{- 9}{16}$

Step 6.2.3

Rewrite the expression.

$\frac{4 a^{2}}{1} + \frac{- 9}{16}$

Step 6.2.4

Divide $4 a^{2}$ by $1$ .

$4 a^{2} + \frac{- 9}{16}$

Step 7

Move the negative in front of the fraction.

$4 a^{2} - \frac{9}{16}$