Calculus Examples

${(- \frac{2}{3})}^{3} - 5 {(- \frac{2}{3})}^{2} - 8 (- \frac{2}{3}) + 3$

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

${(- 1)}^{3} {(\frac{2}{3})}^{3} - 5 {(- \frac{2}{3})}^{2} - 8 (- \frac{2}{3}) + 3$

Step 1.1.2

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

${(- 1)}^{3} \frac{2^{3}}{3^{3}} - 5 {(- \frac{2}{3})}^{2} - 8 (- \frac{2}{3}) + 3$

Step 1.2

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

$- \frac{2^{3}}{3^{3}} - 5 {(- \frac{2}{3})}^{2} - 8 (- \frac{2}{3}) + 3$

Step 1.3

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

$- \frac{8}{3^{3}} - 5 {(- \frac{2}{3})}^{2} - 8 (- \frac{2}{3}) + 3$

Step 1.4

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

$- \frac{8}{27} - 5 {(- \frac{2}{3})}^{2} - 8 (- \frac{2}{3}) + 3$

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

$- \frac{8}{27} - 5 ({(- 1)}^{2} {(\frac{2}{3})}^{2}) - 8 (- \frac{2}{3}) + 3$

Step 1.5.2

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

$- \frac{8}{27} - 5 ({(- 1)}^{2} \frac{2^{2}}{3^{2}}) - 8 (- \frac{2}{3}) + 3$

Step 1.6

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

$- \frac{8}{27} - 5 (1 \frac{2^{2}}{3^{2}}) - 8 (- \frac{2}{3}) + 3$

Step 1.7

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

$- \frac{8}{27} - 5 \frac{2^{2}}{3^{2}} - 8 (- \frac{2}{3}) + 3$

Step 1.8

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

$- \frac{8}{27} - 5 \frac{4}{3^{2}} - 8 (- \frac{2}{3}) + 3$

Step 1.9

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

$- \frac{8}{27} - 5 (\frac{4}{9}) - 8 (- \frac{2}{3}) + 3$

Step 1.10

Multiply $- 5 (\frac{4}{9})$ .

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Step 1.10.1

Combine $- 5$ and $\frac{4}{9}$ .

$- \frac{8}{27} + \frac{- 5 \cdot 4}{9} - 8 (- \frac{2}{3}) + 3$

Step 1.10.2

Multiply $- 5$ by $4$ .

$- \frac{8}{27} + \frac{- 20}{9} - 8 (- \frac{2}{3}) + 3$

Step 1.11

Move the negative in front of the fraction.

$- \frac{8}{27} - \frac{20}{9} - 8 (- \frac{2}{3}) + 3$

Step 1.12

Multiply $- 8 (- \frac{2}{3})$ .

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Step 1.12.1

Multiply $- 1$ by $- 8$ .

$- \frac{8}{27} - \frac{20}{9} + 8 (\frac{2}{3}) + 3$

Step 1.12.2

Combine $8$ and $\frac{2}{3}$ .

$- \frac{8}{27} - \frac{20}{9} + \frac{8 \cdot 2}{3} + 3$

Step 1.12.3

Multiply $8$ by $2$ .

$- \frac{8}{27} - \frac{20}{9} + \frac{16}{3} + 3$

Step 2

Find the common denominator.

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Step 2.1

Multiply $\frac{20}{9}$ by $\frac{3}{3}$ .

$- \frac{8}{27} - (\frac{20}{9} \cdot \frac{3}{3}) + \frac{16}{3} + 3$

Step 2.2

Multiply $\frac{20}{9}$ by $\frac{3}{3}$ .

$- \frac{8}{27} - \frac{20 \cdot 3}{9 \cdot 3} + \frac{16}{3} + 3$

Step 2.3

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

$- \frac{8}{27} - \frac{20 \cdot 3}{9 \cdot 3} + \frac{16}{3} \cdot \frac{9}{9} + 3$

Step 2.4

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

$- \frac{8}{27} - \frac{20 \cdot 3}{9 \cdot 3} + \frac{16 \cdot 9}{3 \cdot 9} + 3$

Step 2.5

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

$- \frac{8}{27} - \frac{20 \cdot 3}{9 \cdot 3} + \frac{16 \cdot 9}{3 \cdot 9} + \frac{3}{1}$

Step 2.6

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

$- \frac{8}{27} - \frac{20 \cdot 3}{9 \cdot 3} + \frac{16 \cdot 9}{3 \cdot 9} + \frac{3}{1} \cdot \frac{27}{27}$

Step 2.7

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

$- \frac{8}{27} - \frac{20 \cdot 3}{9 \cdot 3} + \frac{16 \cdot 9}{3 \cdot 9} + \frac{3 \cdot 27}{27}$

Step 2.8

Reorder the factors of $9 \cdot 3$ .

$- \frac{8}{27} - \frac{20 \cdot 3}{3 \cdot 9} + \frac{16 \cdot 9}{3 \cdot 9} + \frac{3 \cdot 27}{27}$

Step 2.9

Multiply $3$ by $9$ .

$- \frac{8}{27} - \frac{20 \cdot 3}{27} + \frac{16 \cdot 9}{3 \cdot 9} + \frac{3 \cdot 27}{27}$

Step 2.10

Multiply $3$ by $9$ .

$- \frac{8}{27} - \frac{20 \cdot 3}{27} + \frac{16 \cdot 9}{27} + \frac{3 \cdot 27}{27}$

Step 3

Combine the numerators over the common denominator.

$\frac{- 8 - 20 \cdot 3 + 16 \cdot 9 + 3 \cdot 27}{27}$

Step 4

Simplify each term.

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Step 4.1

Multiply $- 20$ by $3$ .

$\frac{- 8 - 60 + 16 \cdot 9 + 3 \cdot 27}{27}$

Step 4.2

Multiply $16$ by $9$ .

$\frac{- 8 - 60 + 144 + 3 \cdot 27}{27}$

Step 4.3

Multiply $3$ by $27$ .

$\frac{- 8 - 60 + 144 + 81}{27}$

Step 5

Simplify by adding and subtracting.

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Step 5.1

Subtract $60$ from $- 8$ .

$\frac{- 68 + 144 + 81}{27}$

Step 5.2

Add $- 68$ and $144$ .

$\frac{76 + 81}{27}$

Step 5.3

Add $76$ and $81$ .

$\frac{157}{27}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{157}{27}$

Decimal Form:

$5.\overline{814}$

Mixed Number Form:

$5 \frac{22}{27}$