Basic Math Examples

$\frac{a^{2} - 64}{4 a - 20} \cdot \frac{a^{2} + 16 a + 64}{a^{2} + 3 a - 40}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $64$ as $8^{2}$ .

$\frac{a^{2} - 8^{2}}{4 a - 20} \cdot \frac{a^{2} + 16 a + 64}{a^{2} + 3 a - 40}$

Step 1.2

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

$\frac{(a + 8) (a - 8)}{4 a - 20} \cdot \frac{a^{2} + 16 a + 64}{a^{2} + 3 a - 40}$

Step 2

Factor $4$ out of $4 a - 20$ .

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Step 2.1

Factor $4$ out of $4 a$ .

$\frac{(a + 8) (a - 8)}{4 (a) - 20} \cdot \frac{a^{2} + 16 a + 64}{a^{2} + 3 a - 40}$

Step 2.2

Factor $4$ out of $- 20$ .

$\frac{(a + 8) (a - 8)}{4 a + 4 \cdot - 5} \cdot \frac{a^{2} + 16 a + 64}{a^{2} + 3 a - 40}$

Step 2.3

Factor $4$ out of $4 a + 4 \cdot - 5$ .

$\frac{(a + 8) (a - 8)}{4 (a - 5)} \cdot \frac{a^{2} + 16 a + 64}{a^{2} + 3 a - 40}$

Step 3

Factor using the perfect square rule.

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Step 3.1

Rewrite $64$ as $8^{2}$ .

$\frac{(a + 8) (a - 8)}{4 (a - 5)} \cdot \frac{a^{2} + 16 a + 8^{2}}{a^{2} + 3 a - 40}$

Step 3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$16 a = 2 \cdot a \cdot 8$

Step 3.3

Rewrite the polynomial.

$\frac{(a + 8) (a - 8)}{4 (a - 5)} \cdot \frac{a^{2} + 2 \cdot a \cdot 8 + 8^{2}}{a^{2} + 3 a - 40}$

Step 3.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = a$ and $b = 8$ .

$\frac{(a + 8) (a - 8)}{4 (a - 5)} \cdot \frac{{(a + 8)}^{2}}{a^{2} + 3 a - 40}$

Step 4

Factor $a^{2} + 3 a - 40$ using the AC method.

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Step 4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 40$ and whose sum is $3$ .

$- 5, 8$

Step 4.2

Write the factored form using these integers.

$\frac{(a + 8) (a - 8)}{4 (a - 5)} \cdot \frac{{(a + 8)}^{2}}{(a - 5) (a + 8)}$

Step 5

Combine.

$\frac{(a + 8) (a - 8) {(a + 8)}^{2}}{4 (a - 5) ((a - 5) (a + 8))}$

Step 6

Multiply $a + 8$ by ${(a + 8)}^{2}$ by adding the exponents.

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Step 6.1

Move ${(a + 8)}^{2}$ .

$\frac{{(a + 8)}^{2} (a + 8) (a - 8)}{4 (a - 5) ((a - 5) (a + 8))}$

Step 6.2

Multiply ${(a + 8)}^{2}$ by $a + 8$ .

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Step 6.2.1

Raise $a + 8$ to the power of $1$ .

$\frac{{(a + 8)}^{2} {(a + 8)}^{1} (a - 8)}{4 (a - 5) ((a - 5) (a + 8))}$

Step 6.2.2

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

$\frac{{(a + 8)}^{2 + 1} (a - 8)}{4 (a - 5) ((a - 5) (a + 8))}$

Step 6.3

Add $2$ and $1$ .

$\frac{{(a + 8)}^{3} (a - 8)}{4 (a - 5) ((a - 5) (a + 8))}$

Step 7

Cancel the common factor of ${(a + 8)}^{3}$ and $a + 8$ .

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Step 7.1

Factor $a + 8$ out of ${(a + 8)}^{3} (a - 8)$ .

$\frac{(a + 8) ({(a + 8)}^{2} (a - 8))}{4 (a - 5) ((a - 5) (a + 8))}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $a + 8$ out of $4 (a - 5) ((a - 5) (a + 8))$ .

$\frac{(a + 8) ({(a + 8)}^{2} (a - 8))}{(a + 8) (4 (a - 5) (a - 5))}$

Step 7.2.2

Cancel the common factor.

$\frac{(a + 8) ({(a + 8)}^{2} (a - 8))}{(a + 8) (4 (a - 5) (a - 5))}$

Step 7.2.3

Rewrite the expression.

$\frac{{(a + 8)}^{2} (a - 8)}{4 (a - 5) (a - 5)}$

Step 8

Simplify the denominator.

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Step 8.1

Raise $a - 5$ to the power of $1$ .

$\frac{{(a + 8)}^{2} (a - 8)}{4 ({(a - 5)}^{1} (a - 5))}$

Step 8.2

Raise $a - 5$ to the power of $1$ .

$\frac{{(a + 8)}^{2} (a - 8)}{4 ({(a - 5)}^{1} {(a - 5)}^{1})}$

Step 8.3

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

$\frac{{(a + 8)}^{2} (a - 8)}{4 {(a - 5)}^{1 + 1}}$

Step 8.4

Add $1$ and $1$ .

$\frac{{(a + 8)}^{2} (a - 8)}{4 {(a - 5)}^{2}}$