Trigonometry Examples

$f (x) = 7 \cdot {(\frac{1}{2})}^{4} - {(\frac{1}{2})}^{3} - 7 \cdot \frac{1}{2} - 1$

Step 1

Apply the product rule to $\frac{1}{2}$ .

$f (x) = 7 \cdot \frac{1^{4}}{2^{4}} - {(\frac{1}{2})}^{3} - 7 \cdot \frac{1}{2} - 1$

Step 2

One to any power is one.

$f (x) = 7 \cdot \frac{1}{2^{4}} - {(\frac{1}{2})}^{3} - 7 \cdot \frac{1}{2} - 1$

Step 3

Raise $2$ to the power of $4$ .

$f (x) = 7 \cdot \frac{1}{16} - {(\frac{1}{2})}^{3} - 7 \cdot \frac{1}{2} - 1$

Step 4

Multiply $7$ by $\frac{1}{16}$ .

$f (x) = 7 (\frac{1}{16}) - {(\frac{1}{2})}^{3} - 7 \cdot \frac{1}{2} - 1$

Step 5

Combine $7$ and $\frac{1}{16}$ .

$f (x) = \frac{7}{16} - {(\frac{1}{2})}^{3} - 7 \cdot \frac{1}{2} - 1$

Step 6

Apply the product rule to $\frac{1}{2}$ .

$f (x) = \frac{7}{16} - \frac{1^{3}}{2^{3}} - 7 \cdot \frac{1}{2} - 1$

Step 7

One to any power is one.

$f (x) = \frac{7}{16} - \frac{1}{2^{3}} - 7 \cdot \frac{1}{2} - 1$

Step 8

Raise $2$ to the power of $3$ .

$f (x) = \frac{7}{16} - \frac{1}{8} - 7 \cdot \frac{1}{2} - 1$

Step 9

Multiply $- 7$ by $\frac{1}{2}$ .

$f (x) = \frac{7}{16} - \frac{1}{8} - 7 (\frac{1}{2}) - 1$

Step 10

Combine $- 7$ and $\frac{1}{2}$ .

$f (x) = \frac{7}{16} - \frac{1}{8} + \frac{- 7}{2} - 1$

Step 11

Move the negative in front of the fraction.

$f (x) = \frac{7}{16} - \frac{1}{8} - \frac{7}{2} - 1$

Step 12

To write $- \frac{1}{8}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (x) = \frac{7}{16} - \frac{1}{8} \cdot \frac{2}{2} - \frac{7}{2} - 1$

Step 13

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 13.1

Multiply $\frac{1}{8}$ by $\frac{2}{2}$ .

$f (x) = \frac{7}{16} - \frac{2}{8 \cdot 2} - \frac{7}{2} - 1$

Step 13.2

Multiply $8$ by $2$ .

$f (x) = \frac{7}{16} - \frac{2}{16} - \frac{7}{2} - 1$

Step 14

Combine the numerators over the common denominator.

$f (x) = \frac{7 - 2}{16} - \frac{7}{2} - 1$

Step 15

Subtract $2$ from $7$ .

$f (x) = \frac{5}{16} - \frac{7}{2} - 1$

Step 16

To write $- \frac{7}{2}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$f (x) = \frac{5}{16} - \frac{7}{2} \cdot \frac{8}{8} - 1$

Step 17

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 17.1

Multiply $\frac{7}{2}$ by $\frac{8}{8}$ .

$f (x) = \frac{5}{16} - \frac{7 \cdot 8}{2 \cdot 8} - 1$

Step 17.2

Multiply $2$ by $8$ .

$f (x) = \frac{5}{16} - \frac{7 \cdot 8}{16} - 1$

Step 18

Combine the numerators over the common denominator.

$f (x) = \frac{5 - 7 \cdot 8}{16} - 1$

Step 19

Simplify the numerator.

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Step 19.1

Multiply $- 7$ by $8$ .

$f (x) = \frac{5 - 56}{16} - 1$

Step 19.2

Subtract $56$ from $5$ .

$f (x) = \frac{- 51}{16} - 1$

Step 20

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$f (x) = \frac{- 51}{16} - 1 \cdot \frac{16}{16}$

Step 21

Combine $- 1$ and $\frac{16}{16}$ .

$f (x) = \frac{- 51}{16} + \frac{- 1 \cdot 16}{16}$

Step 22

Combine the numerators over the common denominator.

$f (x) = \frac{- 51 - 1 \cdot 16}{16}$

Step 23

Simplify the numerator.

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Step 23.1

Multiply $- 1$ by $16$ .

$f (x) = \frac{- 51 - 16}{16}$

Step 23.2

Subtract $16$ from $- 51$ .

$f (x) = \frac{- 67}{16}$

Step 24

Move the negative in front of the fraction.

$f (x) = - \frac{67}{16}$