Calculus Examples

$y = {(- \frac{4}{3})}^{3} - {(- \frac{4}{3})}^{2} - 8 (- \frac{4}{3}) - 3$

Step 1

Simplify each term.

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Step 1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.1.1

Apply the product rule to $- \frac{4}{3}$ .

$y = {(- 1)}^{3} {(\frac{4}{3})}^{3} - {(- \frac{4}{3})}^{2} - 8 (- \frac{4}{3}) - 3$

Step 1.1.2

Apply the product rule to $\frac{4}{3}$ .

$y = {(- 1)}^{3} \frac{4^{3}}{3^{3}} - {(- \frac{4}{3})}^{2} - 8 (- \frac{4}{3}) - 3$

Step 1.2

Raise $- 1$ to the power of $3$ .

$y = - \frac{4^{3}}{3^{3}} - {(- \frac{4}{3})}^{2} - 8 (- \frac{4}{3}) - 3$

Step 1.3

Raise $4$ to the power of $3$ .

$y = - \frac{64}{3^{3}} - {(- \frac{4}{3})}^{2} - 8 (- \frac{4}{3}) - 3$

Step 1.4

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

$y = - \frac{64}{27} - {(- \frac{4}{3})}^{2} - 8 (- \frac{4}{3}) - 3$

Step 1.5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.5.1

Apply the product rule to $- \frac{4}{3}$ .

$y = - \frac{64}{27} - ({(- 1)}^{2} {(\frac{4}{3})}^{2}) - 8 (- \frac{4}{3}) - 3$

Step 1.5.2

Apply the product rule to $\frac{4}{3}$ .

$y = - \frac{64}{27} - ({(- 1)}^{2} \frac{4^{2}}{3^{2}}) - 8 (- \frac{4}{3}) - 3$

Step 1.6

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 1.6.1

Move ${(- 1)}^{2}$ .

$y = - \frac{64}{27} + {(- 1)}^{2} \cdot - 1 \frac{4^{2}}{3^{2}} - 8 (- \frac{4}{3}) - 3$

Step 1.6.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 1.6.2.1

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

$y = - \frac{64}{27} + {(- 1)}^{2} \cdot {(- 1)}^{1} \frac{4^{2}}{3^{2}} - 8 (- \frac{4}{3}) - 3$

Step 1.6.2.2

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

$y = - \frac{64}{27} + {(- 1)}^{2 + 1} \frac{4^{2}}{3^{2}} - 8 (- \frac{4}{3}) - 3$

Step 1.6.3

Add $2$ and $1$ .

$y = - \frac{64}{27} + {(- 1)}^{3} \frac{4^{2}}{3^{2}} - 8 (- \frac{4}{3}) - 3$

Step 1.7

Raise $- 1$ to the power of $3$ .

$y = - \frac{64}{27} - \frac{4^{2}}{3^{2}} - 8 (- \frac{4}{3}) - 3$

Step 1.8

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

$y = - \frac{64}{27} - \frac{16}{3^{2}} - 8 (- \frac{4}{3}) - 3$

Step 1.9

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

$y = - \frac{64}{27} - \frac{16}{9} - 8 (- \frac{4}{3}) - 3$

Step 1.10

Multiply $- 8 (- \frac{4}{3})$ .

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Step 1.10.1

Multiply $- 1$ by $- 8$ .

$y = - \frac{64}{27} - \frac{16}{9} + 8 (\frac{4}{3}) - 3$

Step 1.10.2

Combine $8$ and $\frac{4}{3}$ .

$y = - \frac{64}{27} - \frac{16}{9} + \frac{8 \cdot 4}{3} - 3$

Step 1.10.3

Multiply $8$ by $4$ .

$y = - \frac{64}{27} - \frac{16}{9} + \frac{32}{3} - 3$

Step 2

Find the common denominator.

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Step 2.1

Multiply $\frac{16}{9}$ by $\frac{3}{3}$ .

$y = - \frac{64}{27} - (\frac{16}{9} \cdot \frac{3}{3}) + \frac{32}{3} - 3$

Step 2.2

Multiply $\frac{16}{9}$ by $\frac{3}{3}$ .

$y = - \frac{64}{27} - \frac{16 \cdot 3}{9 \cdot 3} + \frac{32}{3} - 3$

Step 2.3

Multiply $\frac{32}{3}$ by $\frac{9}{9}$ .

$y = - \frac{64}{27} - \frac{16 \cdot 3}{9 \cdot 3} + \frac{32}{3} \cdot \frac{9}{9} - 3$

Step 2.4

Multiply $\frac{32}{3}$ by $\frac{9}{9}$ .

$y = - \frac{64}{27} - \frac{16 \cdot 3}{9 \cdot 3} + \frac{32 \cdot 9}{3 \cdot 9} - 3$

Step 2.5

Write $- 3$ as a fraction with denominator $1$ .

$y = - \frac{64}{27} - \frac{16 \cdot 3}{9 \cdot 3} + \frac{32 \cdot 9}{3 \cdot 9} + \frac{- 3}{1}$

Step 2.6

Multiply $\frac{- 3}{1}$ by $\frac{27}{27}$ .

$y = - \frac{64}{27} - \frac{16 \cdot 3}{9 \cdot 3} + \frac{32 \cdot 9}{3 \cdot 9} + \frac{- 3}{1} \cdot \frac{27}{27}$

Step 2.7

Multiply $\frac{- 3}{1}$ by $\frac{27}{27}$ .

$y = - \frac{64}{27} - \frac{16 \cdot 3}{9 \cdot 3} + \frac{32 \cdot 9}{3 \cdot 9} + \frac{- 3 \cdot 27}{27}$

Step 2.8

Reorder the factors of $9 \cdot 3$ .

$y = - \frac{64}{27} - \frac{16 \cdot 3}{3 \cdot 9} + \frac{32 \cdot 9}{3 \cdot 9} + \frac{- 3 \cdot 27}{27}$

Step 2.9

Multiply $3$ by $9$ .

$y = - \frac{64}{27} - \frac{16 \cdot 3}{27} + \frac{32 \cdot 9}{3 \cdot 9} + \frac{- 3 \cdot 27}{27}$

Step 2.10

Multiply $3$ by $9$ .

$y = - \frac{64}{27} - \frac{16 \cdot 3}{27} + \frac{32 \cdot 9}{27} + \frac{- 3 \cdot 27}{27}$

Step 3

Combine the numerators over the common denominator.

$y = \frac{- 64 - 16 \cdot 3 + 32 \cdot 9 - 3 \cdot 27}{27}$

Step 4

Simplify each term.

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Step 4.1

Multiply $- 16$ by $3$ .

$y = \frac{- 64 - 48 + 32 \cdot 9 - 3 \cdot 27}{27}$

Step 4.2

Multiply $32$ by $9$ .

$y = \frac{- 64 - 48 + 288 - 3 \cdot 27}{27}$

Step 4.3

Multiply $- 3$ by $27$ .

$y = \frac{- 64 - 48 + 288 - 81}{27}$

Step 5

Simplify by adding and subtracting.

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Step 5.1

Subtract $48$ from $- 64$ .

$y = \frac{- 112 + 288 - 81}{27}$

Step 5.2

Add $- 112$ and $288$ .

$y = \frac{176 - 81}{27}$

Step 5.3

Subtract $81$ from $176$ .

$y = \frac{95}{27}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$y = \frac{95}{27}$

Decimal Form:

$y = 3.\overline{518}$

Mixed Number Form:

$y = 3 \frac{14}{27}$