Basic Math Examples

$\frac{\frac{15}{16 - a} + \frac{16}{a - 16}}{\frac{7}{a} + \frac{9}{a - 16}}$

Step 1

Multiply the numerator and denominator of the fraction by $(16 - a) (a - 16) a$ .

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Step 1.1

Multiply $\frac{\frac{15}{16 - a} + \frac{16}{a - 16}}{\frac{7}{a} + \frac{9}{a - 16}}$ by $\frac{(16 - a) (a - 16) a}{(16 - a) (a - 16) a}$ .

$\frac{(16 - a) (a - 16) a}{(16 - a) (a - 16) a} \cdot \frac{\frac{15}{16 - a} + \frac{16}{a - 16}}{\frac{7}{a} + \frac{9}{a - 16}}$

Step 1.2

Combine.

$\frac{(16 - a) (a - 16) a (\frac{15}{16 - a} + \frac{16}{a - 16})}{(16 - a) (a - 16) a (\frac{7}{a} + \frac{9}{a - 16})}$

Step 2

Apply the distributive property.

$\frac{(16 - a) (a - 16) a \frac{15}{16 - a} + (16 - a) (a - 16) a \frac{16}{a - 16}}{(16 - a) (a - 16) a \frac{7}{a} + (16 - a) (a - 16) a \frac{9}{a - 16}}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $16 - a$ .

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Step 3.1.1

Factor $16 - a$ out of $(16 - a) (a - 16) a$ .

$\frac{(16 - a) ((a - 16) a) \frac{15}{16 - a} + (16 - a) (a - 16) a \frac{16}{a - 16}}{(16 - a) (a - 16) a \frac{7}{a} + (16 - a) (a - 16) a \frac{9}{a - 16}}$

Step 3.1.2

Cancel the common factor.

$\frac{(16 - a) ((a - 16) a) \frac{15}{16 - a} + (16 - a) (a - 16) a \frac{16}{a - 16}}{(16 - a) (a - 16) a \frac{7}{a} + (16 - a) (a - 16) a \frac{9}{a - 16}}$

Step 3.1.3

Rewrite the expression.

$\frac{(a - 16) a \cdot 15 + (16 - a) (a - 16) a \frac{16}{a - 16}}{(16 - a) (a - 16) a \frac{7}{a} + (16 - a) (a - 16) a \frac{9}{a - 16}}$

Step 3.2

Cancel the common factor of $a - 16$ .

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Step 3.2.1

Factor $a - 16$ out of $(16 - a) (a - 16) a$ .

$\frac{(a - 16) a \cdot 15 + (a - 16) ((16 - a) a) \frac{16}{a - 16}}{(16 - a) (a - 16) a \frac{7}{a} + (16 - a) (a - 16) a \frac{9}{a - 16}}$

Step 3.2.2

Cancel the common factor.

$\frac{(a - 16) a \cdot 15 + (a - 16) ((16 - a) a) \frac{16}{a - 16}}{(16 - a) (a - 16) a \frac{7}{a} + (16 - a) (a - 16) a \frac{9}{a - 16}}$

Step 3.2.3

Rewrite the expression.

$\frac{(a - 16) a \cdot 15 + (16 - a) a \cdot 16}{(16 - a) (a - 16) a \frac{7}{a} + (16 - a) (a - 16) a \frac{9}{a - 16}}$

Step 3.3

Cancel the common factor of $a$ .

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Step 3.3.1

Factor $a$ out of $(16 - a) (a - 16) a$ .

$\frac{(a - 16) a \cdot 15 + (16 - a) a \cdot 16}{a ((16 - a) (a - 16)) \frac{7}{a} + (16 - a) (a - 16) a \frac{9}{a - 16}}$

Step 3.3.2

Cancel the common factor.

$\frac{(a - 16) a \cdot 15 + (16 - a) a \cdot 16}{a ((16 - a) (a - 16)) \frac{7}{a} + (16 - a) (a - 16) a \frac{9}{a - 16}}$

Step 3.3.3

Rewrite the expression.

$\frac{(a - 16) a \cdot 15 + (16 - a) a \cdot 16}{(16 - a) (a - 16) \cdot 7 + (16 - a) (a - 16) a \frac{9}{a - 16}}$

Step 3.4

Rewrite $16$ as $- 1 (- 16)$ .

$\frac{(a - 16) a \cdot 15 + (16 - a) a \cdot 16}{(- 1 (- 16) - a) (a - 16) \cdot 7 + (16 - a) (a - 16) a \frac{9}{a - 16}}$

Step 3.5

Factor $- 1$ out of $- a$ .

$\frac{(a - 16) a \cdot 15 + (16 - a) a \cdot 16}{(- 1 (- 16) - (a)) (a - 16) \cdot 7 + (16 - a) (a - 16) a \frac{9}{a - 16}}$

Step 3.6

Factor $- 1$ out of $- 1 (- 16) - (a)$ .

$\frac{(a - 16) a \cdot 15 + (16 - a) a \cdot 16}{- 1 (- 16 + a) (a - 16) \cdot 7 + (16 - a) (a - 16) a \frac{9}{a - 16}}$

Step 3.7

Reorder terms.

$\frac{(a - 16) a \cdot 15 + (16 - a) a \cdot 16}{- 1 (a - 16) (a - 16) \cdot 7 + (16 - a) (a - 16) a \frac{9}{a - 16}}$

Step 3.8

Raise $a - 16$ to the power of $1$ .

$\frac{(a - 16) a \cdot 15 + (16 - a) a \cdot 16}{- 1 ({(a - 16)}^{1} (a - 16)) \cdot 7 + (16 - a) (a - 16) a \frac{9}{a - 16}}$

Step 3.9

Raise $a - 16$ to the power of $1$ .

$\frac{(a - 16) a \cdot 15 + (16 - a) a \cdot 16}{- 1 ({(a - 16)}^{1} {(a - 16)}^{1}) \cdot 7 + (16 - a) (a - 16) a \frac{9}{a - 16}}$

Step 3.10

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

$\frac{(a - 16) a \cdot 15 + (16 - a) a \cdot 16}{- 1 {(a - 16)}^{1 + 1} \cdot 7 + (16 - a) (a - 16) a \frac{9}{a - 16}}$

Step 3.11

Add $1$ and $1$ .

$\frac{(a - 16) a \cdot 15 + (16 - a) a \cdot 16}{- 1 {(a - 16)}^{2} \cdot 7 + (16 - a) (a - 16) a \frac{9}{a - 16}}$

Step 3.12

Multiply $7$ by $- 1$ .

$\frac{(a - 16) a \cdot 15 + (16 - a) a \cdot 16}{- 7 {(a - 16)}^{2} + (16 - a) (a - 16) a \frac{9}{a - 16}}$

Step 3.13

Cancel the common factor of $a - 16$ .

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Step 3.13.1

Factor $a - 16$ out of $(16 - a) (a - 16) a$ .

$\frac{(a - 16) a \cdot 15 + (16 - a) a \cdot 16}{- 7 {(a - 16)}^{2} + (a - 16) ((16 - a) a) \frac{9}{a - 16}}$

Step 3.13.2

Cancel the common factor.

$\frac{(a - 16) a \cdot 15 + (16 - a) a \cdot 16}{- 7 {(a - 16)}^{2} + (a - 16) ((16 - a) a) \frac{9}{a - 16}}$

Step 3.13.3

Rewrite the expression.

$\frac{(a - 16) a \cdot 15 + (16 - a) a \cdot 16}{- 7 {(a - 16)}^{2} + (16 - a) a \cdot 9}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $a$ out of $(a - 16) a \cdot 15 + (16 - a) a \cdot 16$ .

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Step 4.1.1

Factor $a$ out of $(a - 16) a \cdot 15$ .

$\frac{a ((a - 16) \cdot 15) + (16 - a) a \cdot 16}{- 7 {(a - 16)}^{2} + (16 - a) a \cdot 9}$

Step 4.1.2

Factor $a$ out of $(16 - a) a \cdot 16$ .

$\frac{a ((a - 16) \cdot 15) + a ((16 - a) \cdot 16)}{- 7 {(a - 16)}^{2} + (16 - a) a \cdot 9}$

Step 4.1.3

Factor $a$ out of $a ((a - 16) \cdot 15) + a ((16 - a) \cdot 16)$ .

$\frac{a ((a - 16) \cdot 15 + (16 - a) \cdot 16)}{- 7 {(a - 16)}^{2} + (16 - a) a \cdot 9}$

Step 4.2

Apply the distributive property.

$\frac{a (a \cdot 15 - 16 \cdot 15 + (16 - a) \cdot 16)}{- 7 {(a - 16)}^{2} + (16 - a) a \cdot 9}$

Step 4.3

Move $15$ to the left of $a$ .

$\frac{a (15 \cdot a - 16 \cdot 15 + (16 - a) \cdot 16)}{- 7 {(a - 16)}^{2} + (16 - a) a \cdot 9}$

Step 4.4

Multiply $- 16$ by $15$ .

$\frac{a (15 \cdot a - 240 + (16 - a) \cdot 16)}{- 7 {(a - 16)}^{2} + (16 - a) a \cdot 9}$

Step 4.5

Apply the distributive property.

$\frac{a (15 a - 240 + 16 \cdot 16 - a \cdot 16)}{- 7 {(a - 16)}^{2} + (16 - a) a \cdot 9}$

Step 4.6

Multiply $16$ by $16$ .

$\frac{a (15 a - 240 + 256 - a \cdot 16)}{- 7 {(a - 16)}^{2} + (16 - a) a \cdot 9}$

Step 4.7

Multiply $16$ by $- 1$ .

$\frac{a (15 a - 240 + 256 - 16 a)}{- 7 {(a - 16)}^{2} + (16 - a) a \cdot 9}$

Step 4.8

Subtract $16 a$ from $15 a$ .

$\frac{a (- a - 240 + 256)}{- 7 {(a - 16)}^{2} + (16 - a) a \cdot 9}$

Step 4.9

Add $- 240$ and $256$ .

$\frac{a (- a + 16)}{- 7 {(a - 16)}^{2} + (16 - a) a \cdot 9}$

Step 5

Simplify the denominator.

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Step 5.1

Rewrite ${(a - 16)}^{2}$ as $(a - 16) (a - 16)$ .

$\frac{a (- a + 16)}{- 7 ((a - 16) (a - 16)) + (16 - a) a \cdot 9}$

Step 5.2

Expand $(a - 16) (a - 16)$ using the FOIL Method.

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Step 5.2.1

Apply the distributive property.

$\frac{a (- a + 16)}{- 7 (a (a - 16) - 16 (a - 16)) + (16 - a) a \cdot 9}$

Step 5.2.2

Apply the distributive property.

$\frac{a (- a + 16)}{- 7 (a \cdot a + a \cdot - 16 - 16 (a - 16)) + (16 - a) a \cdot 9}$

Step 5.2.3

Apply the distributive property.

$\frac{a (- a + 16)}{- 7 (a \cdot a + a \cdot - 16 - 16 a - 16 \cdot - 16) + (16 - a) a \cdot 9}$

Step 5.3

Simplify and combine like terms.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Multiply $a$ by $a$ .

$\frac{a (- a + 16)}{- 7 (a^{2} + a \cdot - 16 - 16 a - 16 \cdot - 16) + (16 - a) a \cdot 9}$

Step 5.3.1.2

Move $- 16$ to the left of $a$ .

$\frac{a (- a + 16)}{- 7 (a^{2} - 16 \cdot a - 16 a - 16 \cdot - 16) + (16 - a) a \cdot 9}$

Step 5.3.1.3

Multiply $- 16$ by $- 16$ .

$\frac{a (- a + 16)}{- 7 (a^{2} - 16 a - 16 a + 256) + (16 - a) a \cdot 9}$

Step 5.3.2

Subtract $16 a$ from $- 16 a$ .

$\frac{a (- a + 16)}{- 7 (a^{2} - 32 a + 256) + (16 - a) a \cdot 9}$

Step 5.4

Apply the distributive property.

$\frac{a (- a + 16)}{- 7 a^{2} - 7 (- 32 a) - 7 \cdot 256 + (16 - a) a \cdot 9}$

Step 5.5

Simplify.

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Step 5.5.1

Multiply $- 32$ by $- 7$ .

$\frac{a (- a + 16)}{- 7 a^{2} + 224 a - 7 \cdot 256 + (16 - a) a \cdot 9}$

Step 5.5.2

Multiply $- 7$ by $256$ .

$\frac{a (- a + 16)}{- 7 a^{2} + 224 a - 1792 + (16 - a) a \cdot 9}$

Step 5.6

Apply the distributive property.

$\frac{a (- a + 16)}{- 7 a^{2} + 224 a - 1792 + (16 a - a \cdot a) \cdot 9}$

Step 5.7

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

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Step 5.7.1

Move $a$ .

$\frac{a (- a + 16)}{- 7 a^{2} + 224 a - 1792 + (16 a - (a \cdot a)) \cdot 9}$

Step 5.7.2

Multiply $a$ by $a$ .

$\frac{a (- a + 16)}{- 7 a^{2} + 224 a - 1792 + (16 a - a^{2}) \cdot 9}$

Step 5.8

Apply the distributive property.

$\frac{a (- a + 16)}{- 7 a^{2} + 224 a - 1792 + 16 a \cdot 9 - a^{2} \cdot 9}$

Step 5.9

Multiply $9$ by $16$ .

$\frac{a (- a + 16)}{- 7 a^{2} + 224 a - 1792 + 144 a - a^{2} \cdot 9}$

Step 5.10

Multiply $9$ by $- 1$ .

$\frac{a (- a + 16)}{- 7 a^{2} + 224 a - 1792 + 144 a - 9 a^{2}}$

Step 5.11

Subtract $9 a^{2}$ from $- 7 a^{2}$ .

$\frac{a (- a + 16)}{- 16 a^{2} + 224 a - 1792 + 144 a}$

Step 5.12

Add $224 a$ and $144 a$ .

$\frac{a (- a + 16)}{- 16 a^{2} + 368 a - 1792}$

Step 5.13

Rewrite $- 16 a^{2} + 368 a - 1792$ in a factored form.

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Step 5.13.1

Factor $16$ out of $- 16 a^{2} + 368 a - 1792$ .

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Step 5.13.1.1

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

$\frac{a (- a + 16)}{16 (- a^{2}) + 368 a - 1792}$

Step 5.13.1.2

Factor $16$ out of $368 a$ .

$\frac{a (- a + 16)}{16 (- a^{2}) + 16 (23 a) - 1792}$

Step 5.13.1.3

Factor $16$ out of $- 1792$ .

$\frac{a (- a + 16)}{16 (- a^{2}) + 16 (23 a) + 16 (- 112)}$

Step 5.13.1.4

Factor $16$ out of $16 (- a^{2}) + 16 (23 a)$ .

$\frac{a (- a + 16)}{16 (- a^{2} + 23 a) + 16 (- 112)}$

Step 5.13.1.5

Factor $16$ out of $16 (- a^{2} + 23 a) + 16 (- 112)$ .

$\frac{a (- a + 16)}{16 (- a^{2} + 23 a - 112)}$

Step 5.13.2

Factor by grouping.

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Step 5.13.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 112 = 112$ and whose sum is $b = 23$ .

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Step 5.13.2.1.1

Factor $23$ out of $23 a$ .

$\frac{a (- a + 16)}{16 (- a^{2} + 23 (a) - 112)}$

Step 5.13.2.1.2

Rewrite $23$ as $7$ plus $16$

$\frac{a (- a + 16)}{16 (- a^{2} + (7 + 16) a - 112)}$

Step 5.13.2.1.3

Apply the distributive property.

$\frac{a (- a + 16)}{16 (- a^{2} + 7 a + 16 a - 112)}$

Step 5.13.2.2

Factor out the greatest common factor from each group.

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Step 5.13.2.2.1

Group the first two terms and the last two terms.

$\frac{a (- a + 16)}{16 ((- a^{2} + 7 a) + 16 a - 112)}$

Step 5.13.2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{a (- a + 16)}{16 (a (- a + 7) - 16 (- a + 7))}$

Step 5.13.2.3

Factor the polynomial by factoring out the greatest common factor, $- a + 7$ .

$\frac{a (- a + 16)}{16 ((- a + 7) (a - 16))}$

$\frac{a (- a + 16)}{16 (- a + 7) (a - 16)}$

Step 6

Simplify terms.

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Step 6.1

Cancel the common factor of $- a + 16$ and $a - 16$ .

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Step 6.1.1

Factor $- 1$ out of $- a$ .

$\frac{a (- (a) + 16)}{16 (- a + 7) (a - 16)}$

Step 6.1.2

Rewrite $16$ as $- 1 (- 16)$ .

$\frac{a (- (a) - 1 (- 16))}{16 (- a + 7) (a - 16)}$

Step 6.1.3

Factor $- 1$ out of $- (a) - 1 (- 16)$ .

$\frac{a (- (a - 16))}{16 (- a + 7) (a - 16)}$

Step 6.1.4

Rewrite $- (a - 16)$ as $- 1 (a - 16)$ .

$\frac{a (- 1 (a - 16))}{16 (- a + 7) (a - 16)}$

Step 6.1.5

Cancel the common factor.

$\frac{a (- 1 (a - 16))}{16 (- a + 7) (a - 16)}$

Step 6.1.6

Rewrite the expression.

$\frac{a \cdot (- 1)}{16 (- a + 7)}$

Step 6.2

Simplify the expression.

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Step 6.2.1

Move $- 1$ to the left of $a$ .

$\frac{- 1 \cdot a}{16 (- a + 7)}$

Step 6.2.2

Move the negative in front of the fraction.

$- \frac{a}{16 (- a + 7)}$

Step 6.3

Factor $- 1$ out of $- a$ .

$- \frac{a}{16 (- (a) + 7)}$

Step 6.4

Rewrite $7$ as $- 1 (- 7)$ .

$- \frac{a}{16 (- (a) - 1 (- 7))}$

Step 6.5

Factor $- 1$ out of $- (a) - 1 (- 7)$ .

$- \frac{a}{16 (- (a - 7))}$

Step 6.6

Simplify the expression.

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Step 6.6.1

Rewrite $- (a - 7)$ as $- 1 (a - 7)$ .

$- \frac{a}{16 (- 1 (a - 7))}$

Step 6.6.2

Move the negative in front of the fraction.

$- - \frac{a}{16 (a - 7)}$

Step 6.6.3

Multiply $- 1$ by $- 1$ .

$1 \frac{a}{16 (a - 7)}$

Step 6.6.4

Multiply $\frac{a}{16 (a - 7)}$ by $1$ .

$\frac{a}{16 (a - 7)}$