Calculus Examples

$y = {(- \frac{2}{3})}^{3} - 2 {(- \frac{2}{3})}^{2} - 4 (- \frac{2}{3}) + 6$

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 $- \frac{2}{3}$ .

$y = {(- 1)}^{3} {(\frac{2}{3})}^{3} - 2 {(- \frac{2}{3})}^{2} - 4 (- \frac{2}{3}) + 6$

Step 1.1.2

Apply the product rule to $\frac{2}{3}$ .

$y = {(- 1)}^{3} \frac{2^{3}}{3^{3}} - 2 {(- \frac{2}{3})}^{2} - 4 (- \frac{2}{3}) + 6$

Step 1.2

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

$y = - \frac{2^{3}}{3^{3}} - 2 {(- \frac{2}{3})}^{2} - 4 (- \frac{2}{3}) + 6$

Step 1.3

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

$y = - \frac{8}{3^{3}} - 2 {(- \frac{2}{3})}^{2} - 4 (- \frac{2}{3}) + 6$

Step 1.4

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

$y = - \frac{8}{27} - 2 {(- \frac{2}{3})}^{2} - 4 (- \frac{2}{3}) + 6$

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 $- \frac{2}{3}$ .

$y = - \frac{8}{27} - 2 ({(- 1)}^{2} {(\frac{2}{3})}^{2}) - 4 (- \frac{2}{3}) + 6$

Step 1.5.2

Apply the product rule to $\frac{2}{3}$ .

$y = - \frac{8}{27} - 2 ({(- 1)}^{2} \frac{2^{2}}{3^{2}}) - 4 (- \frac{2}{3}) + 6$

Step 1.6

Raise $- 1$ to the power of $2$ .

$y = - \frac{8}{27} - 2 (1 \frac{2^{2}}{3^{2}}) - 4 (- \frac{2}{3}) + 6$

Step 1.7

Multiply $\frac{2^{2}}{3^{2}}$ by $1$ .

$y = - \frac{8}{27} - 2 \frac{2^{2}}{3^{2}} - 4 (- \frac{2}{3}) + 6$

Step 1.8

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

$y = - \frac{8}{27} - 2 \frac{4}{3^{2}} - 4 (- \frac{2}{3}) + 6$

Step 1.9

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

$y = - \frac{8}{27} - 2 (\frac{4}{9}) - 4 (- \frac{2}{3}) + 6$

Step 1.10

Multiply $- 2 (\frac{4}{9})$ .

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Step 1.10.1

Combine $- 2$ and $\frac{4}{9}$ .

$y = - \frac{8}{27} + \frac{- 2 \cdot 4}{9} - 4 (- \frac{2}{3}) + 6$

Step 1.10.2

Multiply $- 2$ by $4$ .

$y = - \frac{8}{27} + \frac{- 8}{9} - 4 (- \frac{2}{3}) + 6$

Step 1.11

Move the negative in front of the fraction.

$y = - \frac{8}{27} - \frac{8}{9} - 4 (- \frac{2}{3}) + 6$

Step 1.12

Multiply $- 4 (- \frac{2}{3})$ .

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Step 1.12.1

Multiply $- 1$ by $- 4$ .

$y = - \frac{8}{27} - \frac{8}{9} + 4 (\frac{2}{3}) + 6$

Step 1.12.2

Combine $4$ and $\frac{2}{3}$ .

$y = - \frac{8}{27} - \frac{8}{9} + \frac{4 \cdot 2}{3} + 6$

Step 1.12.3

Multiply $4$ by $2$ .

$y = - \frac{8}{27} - \frac{8}{9} + \frac{8}{3} + 6$

Step 2

Find the common denominator.

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Step 2.1

Multiply $\frac{8}{9}$ by $\frac{3}{3}$ .

$y = - \frac{8}{27} - (\frac{8}{9} \cdot \frac{3}{3}) + \frac{8}{3} + 6$

Step 2.2

Multiply $\frac{8}{9}$ by $\frac{3}{3}$ .

$y = - \frac{8}{27} - \frac{8 \cdot 3}{9 \cdot 3} + \frac{8}{3} + 6$

Step 2.3

Multiply $\frac{8}{3}$ by $\frac{9}{9}$ .

$y = - \frac{8}{27} - \frac{8 \cdot 3}{9 \cdot 3} + \frac{8}{3} \cdot \frac{9}{9} + 6$

Step 2.4

Multiply $\frac{8}{3}$ by $\frac{9}{9}$ .

$y = - \frac{8}{27} - \frac{8 \cdot 3}{9 \cdot 3} + \frac{8 \cdot 9}{3 \cdot 9} + 6$

Step 2.5

Write $6$ as a fraction with denominator $1$ .

$y = - \frac{8}{27} - \frac{8 \cdot 3}{9 \cdot 3} + \frac{8 \cdot 9}{3 \cdot 9} + \frac{6}{1}$

Step 2.6

Multiply $\frac{6}{1}$ by $\frac{27}{27}$ .

$y = - \frac{8}{27} - \frac{8 \cdot 3}{9 \cdot 3} + \frac{8 \cdot 9}{3 \cdot 9} + \frac{6}{1} \cdot \frac{27}{27}$

Step 2.7

Multiply $\frac{6}{1}$ by $\frac{27}{27}$ .

$y = - \frac{8}{27} - \frac{8 \cdot 3}{9 \cdot 3} + \frac{8 \cdot 9}{3 \cdot 9} + \frac{6 \cdot 27}{27}$

Step 2.8

Reorder the factors of $9 \cdot 3$ .

$y = - \frac{8}{27} - \frac{8 \cdot 3}{3 \cdot 9} + \frac{8 \cdot 9}{3 \cdot 9} + \frac{6 \cdot 27}{27}$

Step 2.9

Multiply $3$ by $9$ .

$y = - \frac{8}{27} - \frac{8 \cdot 3}{27} + \frac{8 \cdot 9}{3 \cdot 9} + \frac{6 \cdot 27}{27}$

Step 2.10

Multiply $3$ by $9$ .

$y = - \frac{8}{27} - \frac{8 \cdot 3}{27} + \frac{8 \cdot 9}{27} + \frac{6 \cdot 27}{27}$

Step 3

Combine the numerators over the common denominator.

$y = \frac{- 8 - 8 \cdot 3 + 8 \cdot 9 + 6 \cdot 27}{27}$

Step 4

Simplify each term.

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Step 4.1

Multiply $- 8$ by $3$ .

$y = \frac{- 8 - 24 + 8 \cdot 9 + 6 \cdot 27}{27}$

Step 4.2

Multiply $8$ by $9$ .

$y = \frac{- 8 - 24 + 72 + 6 \cdot 27}{27}$

Step 4.3

Multiply $6$ by $27$ .

$y = \frac{- 8 - 24 + 72 + 162}{27}$

Step 5

Simplify by adding and subtracting.

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Step 5.1

Subtract $24$ from $- 8$ .

$y = \frac{- 32 + 72 + 162}{27}$

Step 5.2

Add $- 32$ and $72$ .

$y = \frac{40 + 162}{27}$

Step 5.3

Add $40$ and $162$ .

$y = \frac{202}{27}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$y = \frac{202}{27}$

Decimal Form:

$y = 7.\overline{481}$

Mixed Number Form:

$y = 7 \frac{13}{27}$