Calculus Examples

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

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 $- \frac{5}{3}$ .

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

Step 1.1.2

Apply the product rule to $\frac{5}{3}$ .

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

Step 1.2

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

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Step 1.2.1

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

$y = {(- 1)}^{3} \cdot - 1 \frac{5^{3}}{3^{3}} - {(- \frac{5}{3})}^{2} + 5 (- \frac{5}{3}) - 3$

Step 1.2.2

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

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Step 1.2.2.1

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

$y = {(- 1)}^{3} \cdot {(- 1)}^{1} \frac{5^{3}}{3^{3}} - {(- \frac{5}{3})}^{2} + 5 (- \frac{5}{3}) - 3$

Step 1.2.2.2

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

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

Step 1.2.3

Add $3$ and $1$ .

$y = {(- 1)}^{4} \frac{5^{3}}{3^{3}} - {(- \frac{5}{3})}^{2} + 5 (- \frac{5}{3}) - 3$

Step 1.3

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

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

Step 1.4

Multiply $\frac{5^{3}}{3^{3}}$ by $1$ .

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

Step 1.5

Raise $5$ to the power of $3$ .

$y = \frac{125}{3^{3}} - {(- \frac{5}{3})}^{2} + 5 (- \frac{5}{3}) - 3$

Step 1.6

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

$y = \frac{125}{27} - {(- \frac{5}{3})}^{2} + 5 (- \frac{5}{3}) - 3$

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 $- \frac{5}{3}$ .

$y = \frac{125}{27} - ({(- 1)}^{2} {(\frac{5}{3})}^{2}) + 5 (- \frac{5}{3}) - 3$

Step 1.7.2

Apply the product rule to $\frac{5}{3}$ .

$y = \frac{125}{27} - ({(- 1)}^{2} \frac{5^{2}}{3^{2}}) + 5 (- \frac{5}{3}) - 3$

Step 1.8

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

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Step 1.8.1

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

$y = \frac{125}{27} + {(- 1)}^{2} \cdot - 1 \frac{5^{2}}{3^{2}} + 5 (- \frac{5}{3}) - 3$

Step 1.8.2

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

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Step 1.8.2.1

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

$y = \frac{125}{27} + {(- 1)}^{2} \cdot {(- 1)}^{1} \frac{5^{2}}{3^{2}} + 5 (- \frac{5}{3}) - 3$

Step 1.8.2.2

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

$y = \frac{125}{27} + {(- 1)}^{2 + 1} \frac{5^{2}}{3^{2}} + 5 (- \frac{5}{3}) - 3$

Step 1.8.3

Add $2$ and $1$ .

$y = \frac{125}{27} + {(- 1)}^{3} \frac{5^{2}}{3^{2}} + 5 (- \frac{5}{3}) - 3$

Step 1.9

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

$y = \frac{125}{27} - \frac{5^{2}}{3^{2}} + 5 (- \frac{5}{3}) - 3$

Step 1.10

Raise $5$ to the power of $2$ .

$y = \frac{125}{27} - \frac{25}{3^{2}} + 5 (- \frac{5}{3}) - 3$

Step 1.11

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

$y = \frac{125}{27} - \frac{25}{9} + 5 (- \frac{5}{3}) - 3$

Step 1.12

Multiply $5 (- \frac{5}{3})$ .

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Step 1.12.1

Multiply $- 1$ by $5$ .

$y = \frac{125}{27} - \frac{25}{9} - 5 (\frac{5}{3}) - 3$

Step 1.12.2

Combine $- 5$ and $\frac{5}{3}$ .

$y = \frac{125}{27} - \frac{25}{9} + \frac{- 5 \cdot 5}{3} - 3$

Step 1.12.3

Multiply $- 5$ by $5$ .

$y = \frac{125}{27} - \frac{25}{9} + \frac{- 25}{3} - 3$

Step 1.13

Move the negative in front of the fraction.

$y = \frac{125}{27} - \frac{25}{9} - \frac{25}{3} - 3$

Step 2

Find the common denominator.

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Step 2.1

Multiply $\frac{25}{9}$ by $\frac{3}{3}$ .

$y = \frac{125}{27} - (\frac{25}{9} \cdot \frac{3}{3}) - \frac{25}{3} - 3$

Step 2.2

Multiply $\frac{25}{9}$ by $\frac{3}{3}$ .

$y = \frac{125}{27} - \frac{25 \cdot 3}{9 \cdot 3} - \frac{25}{3} - 3$

Step 2.3

Multiply $\frac{25}{3}$ by $\frac{9}{9}$ .

$y = \frac{125}{27} - \frac{25 \cdot 3}{9 \cdot 3} - (\frac{25}{3} \cdot \frac{9}{9}) - 3$

Step 2.4

Multiply $\frac{25}{3}$ by $\frac{9}{9}$ .

$y = \frac{125}{27} - \frac{25 \cdot 3}{9 \cdot 3} - \frac{25 \cdot 9}{3 \cdot 9} - 3$

Step 2.5

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

$y = \frac{125}{27} - \frac{25 \cdot 3}{9 \cdot 3} - \frac{25 \cdot 9}{3 \cdot 9} + \frac{- 3}{1}$

Step 2.6

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

$y = \frac{125}{27} - \frac{25 \cdot 3}{9 \cdot 3} - \frac{25 \cdot 9}{3 \cdot 9} + \frac{- 3}{1} \cdot \frac{27}{27}$

Step 2.7

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

$y = \frac{125}{27} - \frac{25 \cdot 3}{9 \cdot 3} - \frac{25 \cdot 9}{3 \cdot 9} + \frac{- 3 \cdot 27}{27}$

Step 2.8

Reorder the factors of $9 \cdot 3$ .

$y = \frac{125}{27} - \frac{25 \cdot 3}{3 \cdot 9} - \frac{25 \cdot 9}{3 \cdot 9} + \frac{- 3 \cdot 27}{27}$

Step 2.9

Multiply $3$ by $9$ .

$y = \frac{125}{27} - \frac{25 \cdot 3}{27} - \frac{25 \cdot 9}{3 \cdot 9} + \frac{- 3 \cdot 27}{27}$

Step 2.10

Multiply $3$ by $9$ .

$y = \frac{125}{27} - \frac{25 \cdot 3}{27} - \frac{25 \cdot 9}{27} + \frac{- 3 \cdot 27}{27}$

Step 3

Combine the numerators over the common denominator.

$y = \frac{125 - 25 \cdot 3 - 25 \cdot 9 - 3 \cdot 27}{27}$

Step 4

Simplify each term.

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Step 4.1

Multiply $- 25$ by $3$ .

$y = \frac{125 - 75 - 25 \cdot 9 - 3 \cdot 27}{27}$

Step 4.2

Multiply $- 25$ by $9$ .

$y = \frac{125 - 75 - 225 - 3 \cdot 27}{27}$

Step 4.3

Multiply $- 3$ by $27$ .

$y = \frac{125 - 75 - 225 - 81}{27}$

Step 5

Simplify the expression.

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Step 5.1

Subtract $75$ from $125$ .

$y = \frac{50 - 225 - 81}{27}$

Step 5.2

Subtract $225$ from $50$ .

$y = \frac{- 175 - 81}{27}$

Step 5.3

Subtract $81$ from $- 175$ .

$y = \frac{- 256}{27}$

Step 5.4

Move the negative in front of the fraction.

$y = - \frac{256}{27}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$y = - \frac{256}{27}$

Decimal Form:

$y = - 9.\overline{481}$

Mixed Number Form:

$y = - 9 \frac{13}{27}$