Calculus Examples

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

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 $- \frac{4}{3}$ .

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

Step 1.1.2

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

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

Step 1.2

Multiply $- 1$ by ${(- 1)}^{3}$ by adding the exponents.

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Step 1.2.1

Move ${(- 1)}^{3}$ .

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

Step 1.2.2

Multiply ${(- 1)}^{3}$ by $- 1$ .

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Step 1.2.2.1

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

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

Step 1.2.2.2

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

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

Step 1.2.3

Add $3$ and $1$ .

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

Step 1.3

Raise $- 1$ to the power of $4$ .

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

Step 1.4

Multiply $\frac{4^{3}}{3^{3}}$ by $1$ .

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

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

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Step 1.7.1

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

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

Step 1.7.2

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

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

Step 1.8

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

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

Step 1.9

Multiply $\frac{4^{2}}{3^{2}}$ by $1$ .

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

Step 1.10

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

$\frac{64}{27} + \frac{16}{3^{2}} + 8 (- \frac{4}{3}) + 1$

Step 1.11

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

$\frac{64}{27} + \frac{16}{9} + 8 (- \frac{4}{3}) + 1$

Step 1.12

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

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Step 1.12.1

Multiply $- 1$ by $8$ .

$\frac{64}{27} + \frac{16}{9} - 8 (\frac{4}{3}) + 1$

Step 1.12.2

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

$\frac{64}{27} + \frac{16}{9} + \frac{- 8 \cdot 4}{3} + 1$

Step 1.12.3

Multiply $- 8$ by $4$ .

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

Step 1.13

Move the negative in front of the fraction.

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

Step 2

Find the common denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

$\frac{64}{27} + \frac{16 \cdot 3}{9 \cdot 3} - (\frac{32}{3} \cdot \frac{9}{9}) + 1$

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Reorder the factors of $9 \cdot 3$ .

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

Step 2.9

Multiply $3$ by $9$ .

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

Step 2.10

Multiply $3$ by $9$ .

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Simplify each term.

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Step 4.1

Multiply $16$ by $3$ .

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

Step 4.2

Multiply $- 32$ by $9$ .

$\frac{64 + 48 - 288 + 27}{27}$

Step 5

Simplify the expression.

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Step 5.1

Add $64$ and $48$ .

$\frac{112 - 288 + 27}{27}$

Step 5.2

Subtract $288$ from $112$ .

$\frac{- 176 + 27}{27}$

Step 5.3

Add $- 176$ and $27$ .

$\frac{- 149}{27}$

Step 5.4

Move the negative in front of the fraction.

$- \frac{149}{27}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{149}{27}$

Decimal Form:

$- 5.\overline{518}$

Mixed Number Form:

$- 5 \frac{14}{27}$