Calculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

$\frac{81}{2^{4}} - 2 {(\frac{3}{2})}^{3}$

Step 1.3

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

$\frac{81}{16} - 2 {(\frac{3}{2})}^{3}$

Step 1.4

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

$\frac{81}{16} - 2 \frac{3^{3}}{2^{3}}$

Step 1.5

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

$\frac{81}{16} - 2 \frac{27}{2^{3}}$

Step 1.6

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

$\frac{81}{16} - 2 (\frac{27}{8})$

Step 1.7

Cancel the common factor of $2$ .

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Step 1.7.1

Factor $2$ out of $- 2$ .

$\frac{81}{16} + 2 (- 1) \frac{27}{8}$

Step 1.7.2

Factor $2$ out of $8$ .

$\frac{81}{16} + 2 \cdot - 1 \frac{27}{2 \cdot 4}$

Step 1.7.3

Cancel the common factor.

$\frac{81}{16} + 2 \cdot - 1 \frac{27}{2 \cdot 4}$

Step 1.7.4

Rewrite the expression.

$\frac{81}{16} - 1 (\frac{27}{4})$

Step 1.8

Rewrite $- 1 (\frac{27}{4})$ as $- (\frac{27}{4})$ .

$\frac{81}{16} - \frac{27}{4}$

Step 2

To write $- \frac{27}{4}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{81}{16} - \frac{27}{4} \cdot \frac{4}{4}$

Step 3

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{27}{4}$ by $\frac{4}{4}$ .

$\frac{81}{16} - \frac{27 \cdot 4}{4 \cdot 4}$

Step 3.2

Multiply $4$ by $4$ .

$\frac{81}{16} - \frac{27 \cdot 4}{16}$

Step 4

Combine the numerators over the common denominator.

$\frac{81 - 27 \cdot 4}{16}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $- 27$ by $4$ .

$\frac{81 - 108}{16}$

Step 5.2

Subtract $108$ from $81$ .

$\frac{- 27}{16}$

Step 6

Move the negative in front of the fraction.

$- \frac{27}{16}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{27}{16}$

Decimal Form:

$- 1.6875$

Mixed Number Form:

$- 1 \frac{11}{16}$