Calculus Examples

$\frac{64}{(8) (32 - 3 \cdot 8)} + \frac{192 \ln (8)}{{(32 - 3 \cdot 8)}^{2}}$

Step 1

Cancel the common factor of $64$ and $8$ .

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Step 1.1

Factor $8$ out of $64$ .

$\frac{8 \cdot 8}{8 (32 - 3 \cdot 8)} + \frac{192 \ln (8)}{{(32 - 3 \cdot 8)}^{2}}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Cancel the common factor.

$\frac{8 \cdot 8}{8 (32 - 3 \cdot 8)} + \frac{192 \ln (8)}{{(32 - 3 \cdot 8)}^{2}}$

Step 1.2.2

Rewrite the expression.

$\frac{8}{32 - 3 \cdot 8} + \frac{192 \ln (8)}{{(32 - 3 \cdot 8)}^{2}}$

Step 2

Cancel the common factor of $8$ and $32 - 3 \cdot 8$ .

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Step 2.1

Rewrite $32$ as $- 1 (- 32)$ .

$\frac{8}{- 1 (- 32) - 3 \cdot 8} + \frac{192 \ln (8)}{{(32 - 3 \cdot 8)}^{2}}$

Step 2.2

Factor $- 1$ out of $- 3 \cdot 8$ .

$\frac{8}{- 1 (- 32) - (3 \cdot 8)} + \frac{192 \ln (8)}{{(32 - 3 \cdot 8)}^{2}}$

Step 2.3

Factor $- 1$ out of $- 1 (- 32) - (3 \cdot 8)$ .

$\frac{8}{- 1 (- 32 + 3 \cdot 8)} + \frac{192 \ln (8)}{{(32 - 3 \cdot 8)}^{2}}$

Step 2.4

Reorder terms.

$\frac{8}{- 1 (3 \cdot 8 - 32)} + \frac{192 \ln (8)}{{(32 - 3 \cdot 8)}^{2}}$

Step 2.5

Factor $8$ out of $8$ .

$\frac{8 \cdot 1}{- 1 (3 \cdot 8 - 32)} + \frac{192 \ln (8)}{{(32 - 3 \cdot 8)}^{2}}$

Step 2.6

Cancel the common factors.

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Step 2.6.1

Factor $8$ out of $- 1 (3 \cdot 8 - 32)$ .

$\frac{8 \cdot 1}{8 (- 1 (3 - 4))} + \frac{192 \ln (8)}{{(32 - 3 \cdot 8)}^{2}}$

Step 2.6.2

Cancel the common factor.

$\frac{8 \cdot 1}{8 (- 1 (3 - 4))} + \frac{192 \ln (8)}{{(32 - 3 \cdot 8)}^{2}}$

Step 2.6.3

Rewrite the expression.

$\frac{1}{- 1 (3 - 4)} + \frac{192 \ln (8)}{{(32 - 3 \cdot 8)}^{2}}$

Step 3

Cancel the common factor of $1$ and $- 1$ .

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Step 3.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{- 1 (3 - 4)} + \frac{192 \ln (8)}{{(32 - 3 \cdot 8)}^{2}}$

Step 3.2

Move the negative in front of the fraction.

$- \frac{1}{3 - 4} + \frac{192 \ln (8)}{{(32 - 3 \cdot 8)}^{2}}$

Step 4

Subtract $4$ from $3$ .

$- \frac{1}{- 1} + \frac{192 \ln (8)}{{(32 - 3 \cdot 8)}^{2}}$

Step 5

Divide $1$ by $- 1$ .

$- - 1 + \frac{192 \ln (8)}{{(32 - 3 \cdot 8)}^{2}}$

Step 6

Multiply $- 1$ by $- 1$ .

$1 + \frac{192 \ln (8)}{{(32 - 3 \cdot 8)}^{2}}$

Step 7

Simplify $192 \ln (8)$ by moving $192$ inside the logarithm.

$1 + \frac{\ln (8^{192})}{{(32 - 3 \cdot 8)}^{2}}$

Step 8

Simplify the denominator.

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Step 8.1

Multiply $- 3$ by $8$ .

$1 + \frac{\ln (8^{192})}{{(32 - 24)}^{2}}$

Step 8.2

Subtract $24$ from $32$ .

$1 + \frac{\ln (8^{192})}{8^{2}}$

Step 8.3

Raise $8$ to the power of $2$ .

$1 + \frac{\ln (8^{192})}{64}$

Step 9

Expand $\ln (8^{192})$ by moving $192$ outside the logarithm.

$1 + \frac{192 \ln (8)}{64}$

Step 10

Cancel the common factor of $192$ and $64$ .

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Step 10.1

Factor $64$ out of $192 \ln (8)$ .

$1 + \frac{64 (3 \ln (8))}{64}$

Step 10.2

Cancel the common factors.

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Step 10.2.1

Factor $64$ out of $64$ .

$1 + \frac{64 (3 \ln (8))}{64 (1)}$

Step 10.2.2

Cancel the common factor.

$1 + \frac{64 (3 \ln (8))}{64 \cdot 1}$

Step 10.2.3

Rewrite the expression.

$1 + \frac{3 \ln (8)}{1}$

Step 10.2.4

Divide $3 \ln (8)$ by $1$ .

$1 + 3 \ln (8)$

Step 11

Simplify $3 \ln (8)$ by moving $3$ inside the logarithm.

$1 + \ln (8^{3})$

Step 12

Raise $8$ to the power of $3$ .

$1 + \ln (512)$

Step 13

The result can be shown in multiple forms.

Exact Form:

$1 + \ln (512)$

Decimal Form:

$7.23832462 \dots$