Trigonometry Examples

$\frac{\frac{8}{y - 7}}{\frac{9}{y - 8}} + \frac{\frac{5}{y - 8}}{\frac{4}{y - 7}}$

Step 1

Simplify each term.

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Step 1.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{8}{y - 7} \cdot \frac{y - 8}{9} + \frac{\frac{5}{y - 8}}{\frac{4}{y - 7}}$

Step 1.2

Multiply $\frac{8}{y - 7}$ by $\frac{y - 8}{9}$ .

$\frac{8 (y - 8)}{(y - 7) \cdot 9} + \frac{\frac{5}{y - 8}}{\frac{4}{y - 7}}$

Step 1.3

Move $9$ to the left of $y - 7$ .

$\frac{8 (y - 8)}{9 \cdot (y - 7)} + \frac{\frac{5}{y - 8}}{\frac{4}{y - 7}}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{8 (y - 8)}{9 (y - 7)} + \frac{5}{y - 8} \cdot \frac{y - 7}{4}$

Step 1.5

Multiply $\frac{5}{y - 8}$ by $\frac{y - 7}{4}$ .

$\frac{8 (y - 8)}{9 (y - 7)} + \frac{5 (y - 7)}{(y - 8) \cdot 4}$

Step 1.6

Move $4$ to the left of $y - 8$ .

$\frac{8 (y - 8)}{9 (y - 7)} + \frac{5 (y - 7)}{4 (y - 8)}$

Step 2

To write $\frac{8 (y - 8)}{9 (y - 7)}$ as a fraction with a common denominator, multiply by $\frac{4 (y - 8)}{4 (y - 8)}$ .

$\frac{8 (y - 8)}{9 (y - 7)} \cdot \frac{4 (y - 8)}{4 (y - 8)} + \frac{5 (y - 7)}{4 (y - 8)}$

Step 3

To write $\frac{5 (y - 7)}{4 (y - 8)}$ as a fraction with a common denominator, multiply by $\frac{9 (y - 7)}{9 (y - 7)}$ .

$\frac{8 (y - 8)}{9 (y - 7)} \cdot \frac{4 (y - 8)}{4 (y - 8)} + \frac{5 (y - 7)}{4 (y - 8)} \cdot \frac{9 (y - 7)}{9 (y - 7)}$

Step 4

Write each expression with a common denominator of $36 (y - 7) (y - 8)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{8 (y - 8)}{9 (y - 7)}$ by $\frac{4 (y - 8)}{4 (y - 8)}$ .

$\frac{8 (y - 8) (4 (y - 8))}{9 (y - 7) (4 (y - 8))} + \frac{5 (y - 7)}{4 (y - 8)} \cdot \frac{9 (y - 7)}{9 (y - 7)}$

Step 4.2

Multiply $4$ by $9$ .

$\frac{8 (y - 8) (4 (y - 8))}{36 (y - 7) (y - 8)} + \frac{5 (y - 7)}{4 (y - 8)} \cdot \frac{9 (y - 7)}{9 (y - 7)}$

Step 4.3

Multiply $\frac{5 (y - 7)}{4 (y - 8)}$ by $\frac{9 (y - 7)}{9 (y - 7)}$ .

$\frac{8 (y - 8) (4 (y - 8))}{36 (y - 7) (y - 8)} + \frac{5 (y - 7) (9 (y - 7))}{4 (y - 8) (9 (y - 7))}$

Step 4.4

Multiply $9$ by $4$ .

$\frac{8 (y - 8) (4 (y - 8))}{36 (y - 7) (y - 8)} + \frac{5 (y - 7) (9 (y - 7))}{36 (y - 8) (y - 7)}$

Step 4.5

Reorder the factors of $36 (y - 8) (y - 7)$ .

$\frac{8 (y - 8) (4 (y - 8))}{36 (y - 7) (y - 8)} + \frac{5 (y - 7) (9 (y - 7))}{36 (y - 7) (y - 8)}$

Step 5

Combine the numerators over the common denominator.

$\frac{8 (y - 8) (4 (y - 8)) + 5 (y - 7) (9 (y - 7))}{36 (y - 7) (y - 8)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{(8 y + 8 \cdot - 8) (4 (y - 8)) + 5 (y - 7) (9 (y - 7))}{36 (y - 7) (y - 8)}$

Step 6.2

Multiply $8$ by $- 8$ .

$\frac{(8 y - 64) (4 (y - 8)) + 5 (y - 7) (9 (y - 7))}{36 (y - 7) (y - 8)}$

Step 6.3

Apply the distributive property.

$\frac{(8 y - 64) (4 y + 4 \cdot - 8) + 5 (y - 7) (9 (y - 7))}{36 (y - 7) (y - 8)}$

Step 6.4

Multiply $4$ by $- 8$ .

$\frac{(8 y - 64) (4 y - 32) + 5 (y - 7) (9 (y - 7))}{36 (y - 7) (y - 8)}$

Step 6.5

Expand $(8 y - 64) (4 y - 32)$ using the FOIL Method.

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Step 6.5.1

Apply the distributive property.

$\frac{8 y (4 y - 32) - 64 (4 y - 32) + 5 (y - 7) (9 (y - 7))}{36 (y - 7) (y - 8)}$

Step 6.5.2

Apply the distributive property.

$\frac{8 y (4 y) + 8 y \cdot - 32 - 64 (4 y - 32) + 5 (y - 7) (9 (y - 7))}{36 (y - 7) (y - 8)}$

Step 6.5.3

Apply the distributive property.

$\frac{8 y (4 y) + 8 y \cdot - 32 - 64 (4 y) - 64 \cdot - 32 + 5 (y - 7) (9 (y - 7))}{36 (y - 7) (y - 8)}$

Step 6.6

Simplify and combine like terms.

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Step 6.6.1

Simplify each term.

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Step 6.6.1.1

Rewrite using the commutative property of multiplication.

$\frac{8 \cdot 4 y \cdot y + 8 y \cdot - 32 - 64 (4 y) - 64 \cdot - 32 + 5 (y - 7) (9 (y - 7))}{36 (y - 7) (y - 8)}$

Step 6.6.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 6.6.1.2.1

Move $y$ .

$\frac{8 \cdot 4 (y \cdot y) + 8 y \cdot - 32 - 64 (4 y) - 64 \cdot - 32 + 5 (y - 7) (9 (y - 7))}{36 (y - 7) (y - 8)}$

Step 6.6.1.2.2

Multiply $y$ by $y$ .

$\frac{8 \cdot 4 y^{2} + 8 y \cdot - 32 - 64 (4 y) - 64 \cdot - 32 + 5 (y - 7) (9 (y - 7))}{36 (y - 7) (y - 8)}$

Step 6.6.1.3

Multiply $8$ by $4$ .

$\frac{32 y^{2} + 8 y \cdot - 32 - 64 (4 y) - 64 \cdot - 32 + 5 (y - 7) (9 (y - 7))}{36 (y - 7) (y - 8)}$

Step 6.6.1.4

Multiply $- 32$ by $8$ .

$\frac{32 y^{2} - 256 y - 64 (4 y) - 64 \cdot - 32 + 5 (y - 7) (9 (y - 7))}{36 (y - 7) (y - 8)}$

Step 6.6.1.5

Multiply $4$ by $- 64$ .

$\frac{32 y^{2} - 256 y - 256 y - 64 \cdot - 32 + 5 (y - 7) (9 (y - 7))}{36 (y - 7) (y - 8)}$

Step 6.6.1.6

Multiply $- 64$ by $- 32$ .

$\frac{32 y^{2} - 256 y - 256 y + 2048 + 5 (y - 7) (9 (y - 7))}{36 (y - 7) (y - 8)}$

Step 6.6.2

Subtract $256 y$ from $- 256 y$ .

$\frac{32 y^{2} - 512 y + 2048 + 5 (y - 7) (9 (y - 7))}{36 (y - 7) (y - 8)}$

Step 6.7

Apply the distributive property.

$\frac{32 y^{2} - 512 y + 2048 + (5 y + 5 \cdot - 7) (9 (y - 7))}{36 (y - 7) (y - 8)}$

Step 6.8

Multiply $5$ by $- 7$ .

$\frac{32 y^{2} - 512 y + 2048 + (5 y - 35) (9 (y - 7))}{36 (y - 7) (y - 8)}$

Step 6.9

Apply the distributive property.

$\frac{32 y^{2} - 512 y + 2048 + (5 y - 35) (9 y + 9 \cdot - 7)}{36 (y - 7) (y - 8)}$

Step 6.10

Multiply $9$ by $- 7$ .

$\frac{32 y^{2} - 512 y + 2048 + (5 y - 35) (9 y - 63)}{36 (y - 7) (y - 8)}$

Step 6.11

Expand $(5 y - 35) (9 y - 63)$ using the FOIL Method.

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Step 6.11.1

Apply the distributive property.

$\frac{32 y^{2} - 512 y + 2048 + 5 y (9 y - 63) - 35 (9 y - 63)}{36 (y - 7) (y - 8)}$

Step 6.11.2

Apply the distributive property.

$\frac{32 y^{2} - 512 y + 2048 + 5 y (9 y) + 5 y \cdot - 63 - 35 (9 y - 63)}{36 (y - 7) (y - 8)}$

Step 6.11.3

Apply the distributive property.

$\frac{32 y^{2} - 512 y + 2048 + 5 y (9 y) + 5 y \cdot - 63 - 35 (9 y) - 35 \cdot - 63}{36 (y - 7) (y - 8)}$

Step 6.12

Simplify and combine like terms.

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Step 6.12.1

Simplify each term.

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Step 6.12.1.1

Rewrite using the commutative property of multiplication.

$\frac{32 y^{2} - 512 y + 2048 + 5 \cdot 9 y \cdot y + 5 y \cdot - 63 - 35 (9 y) - 35 \cdot - 63}{36 (y - 7) (y - 8)}$

Step 6.12.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 6.12.1.2.1

Move $y$ .

$\frac{32 y^{2} - 512 y + 2048 + 5 \cdot 9 (y \cdot y) + 5 y \cdot - 63 - 35 (9 y) - 35 \cdot - 63}{36 (y - 7) (y - 8)}$

Step 6.12.1.2.2

Multiply $y$ by $y$ .

$\frac{32 y^{2} - 512 y + 2048 + 5 \cdot 9 y^{2} + 5 y \cdot - 63 - 35 (9 y) - 35 \cdot - 63}{36 (y - 7) (y - 8)}$

Step 6.12.1.3

Multiply $5$ by $9$ .

$\frac{32 y^{2} - 512 y + 2048 + 45 y^{2} + 5 y \cdot - 63 - 35 (9 y) - 35 \cdot - 63}{36 (y - 7) (y - 8)}$

Step 6.12.1.4

Multiply $- 63$ by $5$ .

$\frac{32 y^{2} - 512 y + 2048 + 45 y^{2} - 315 y - 35 (9 y) - 35 \cdot - 63}{36 (y - 7) (y - 8)}$

Step 6.12.1.5

Multiply $9$ by $- 35$ .

$\frac{32 y^{2} - 512 y + 2048 + 45 y^{2} - 315 y - 315 y - 35 \cdot - 63}{36 (y - 7) (y - 8)}$

Step 6.12.1.6

Multiply $- 35$ by $- 63$ .

$\frac{32 y^{2} - 512 y + 2048 + 45 y^{2} - 315 y - 315 y + 2205}{36 (y - 7) (y - 8)}$

Step 6.12.2

Subtract $315 y$ from $- 315 y$ .

$\frac{32 y^{2} - 512 y + 2048 + 45 y^{2} - 630 y + 2205}{36 (y - 7) (y - 8)}$

Step 6.13

Add $32 y^{2}$ and $45 y^{2}$ .

$\frac{77 y^{2} - 512 y + 2048 - 630 y + 2205}{36 (y - 7) (y - 8)}$

Step 6.14

Subtract $630 y$ from $- 512 y$ .

$\frac{77 y^{2} - 1142 y + 2048 + 2205}{36 (y - 7) (y - 8)}$

Step 6.15

Add $2048$ and $2205$ .

$\frac{77 y^{2} - 1142 y + 4253}{36 (y - 7) (y - 8)}$