Basic Math Examples

$18 e^{- \frac{25}{60}} - 12 e^{\frac{- 25 - 20}{60}}$

Step 1

Simplify each term.

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Step 1.1

Cancel the common factor of $25$ and $60$ .

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Step 1.1.1

Factor $5$ out of $25$ .

$18 e^{- \frac{5 (5)}{60}} - 12 e^{\frac{- 25 - 20}{60}}$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $5$ out of $60$ .

$18 e^{- \frac{5 \cdot 5}{5 \cdot 12}} - 12 e^{\frac{- 25 - 20}{60}}$

Step 1.1.2.2

Cancel the common factor.

$18 e^{- \frac{5 \cdot 5}{5 \cdot 12}} - 12 e^{\frac{- 25 - 20}{60}}$

Step 1.1.2.3

Rewrite the expression.

$18 e^{- \frac{5}{12}} - 12 e^{\frac{- 25 - 20}{60}}$

Step 1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$18 \frac{1}{e^{\frac{5}{12}}} - 12 e^{\frac{- 25 - 20}{60}}$

Step 1.3

Combine $18$ and $\frac{1}{e^{\frac{5}{12}}}$ .

$\frac{18}{e^{\frac{5}{12}}} - 12 e^{\frac{- 25 - 20}{60}}$

Step 1.4

Cancel the common factor of $- 25 - 20$ and $60$ .

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Step 1.4.1

Factor $5$ out of $- 25$ .

$\frac{18}{e^{\frac{5}{12}}} - 12 e^{\frac{5 (- 5) - 20}{60}}$

Step 1.4.2

Factor $5$ out of $- 20$ .

$\frac{18}{e^{\frac{5}{12}}} - 12 e^{\frac{5 \cdot - 5 + 5 \cdot - 4}{60}}$

Step 1.4.3

Factor $5$ out of $5 \cdot - 5 + 5 \cdot - 4$ .

$\frac{18}{e^{\frac{5}{12}}} - 12 e^{\frac{5 \cdot (- 5 - 4)}{60}}$

Step 1.4.4

Cancel the common factors.

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Step 1.4.4.1

Factor $5$ out of $60$ .

$\frac{18}{e^{\frac{5}{12}}} - 12 e^{\frac{5 \cdot (- 5 - 4)}{5 (12)}}$

Step 1.4.4.2

Cancel the common factor.

$\frac{18}{e^{\frac{5}{12}}} - 12 e^{\frac{5 \cdot (- 5 - 4)}{5 \cdot 12}}$

Step 1.4.4.3

Rewrite the expression.

$\frac{18}{e^{\frac{5}{12}}} - 12 e^{\frac{- 5 - 4}{12}}$

Step 1.5

Subtract $4$ from $- 5$ .

$\frac{18}{e^{\frac{5}{12}}} - 12 e^{\frac{- 9}{12}}$

Step 1.6

Cancel the common factor of $- 9$ and $12$ .

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Step 1.6.1

Factor $3$ out of $- 9$ .

$\frac{18}{e^{\frac{5}{12}}} - 12 e^{\frac{3 (- 3)}{12}}$

Step 1.6.2

Cancel the common factors.

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Step 1.6.2.1

Factor $3$ out of $12$ .

$\frac{18}{e^{\frac{5}{12}}} - 12 e^{\frac{3 \cdot - 3}{3 \cdot 4}}$

Step 1.6.2.2

Cancel the common factor.

$\frac{18}{e^{\frac{5}{12}}} - 12 e^{\frac{3 \cdot - 3}{3 \cdot 4}}$

Step 1.6.2.3

Rewrite the expression.

$\frac{18}{e^{\frac{5}{12}}} - 12 e^{\frac{- 3}{4}}$

Step 1.7

Move the negative in front of the fraction.

$\frac{18}{e^{\frac{5}{12}}} - 12 e^{- \frac{3}{4}}$

Step 1.8

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{18}{e^{\frac{5}{12}}} - 12 \frac{1}{e^{\frac{3}{4}}}$

Step 1.9

Combine $- 12$ and $\frac{1}{e^{\frac{3}{4}}}$ .

$\frac{18}{e^{\frac{5}{12}}} + \frac{- 12}{e^{\frac{3}{4}}}$

Step 1.10

Move the negative in front of the fraction.

$\frac{18}{e^{\frac{5}{12}}} - \frac{12}{e^{\frac{3}{4}}}$

Step 2

To write $\frac{18}{e^{\frac{5}{12}}}$ as a fraction with a common denominator, multiply by $\frac{e^{\frac{4}{12}}}{e^{\frac{4}{12}}}$ .

$\frac{18}{e^{\frac{5}{12}}} \cdot \frac{e^{\frac{4}{12}}}{e^{\frac{4}{12}}} - \frac{12}{e^{\frac{3}{4}}}$

Step 3

Write each expression with a common denominator of $e^{\frac{3}{4}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{18}{e^{\frac{5}{12}}}$ by $\frac{e^{\frac{4}{12}}}{e^{\frac{4}{12}}}$ .

$\frac{18 e^{\frac{4}{12}}}{e^{\frac{5}{12}} e^{\frac{4}{12}}} - \frac{12}{e^{\frac{3}{4}}}$

Step 3.2

Multiply $e^{\frac{5}{12}}$ by $e^{\frac{4}{12}}$ by adding the exponents.

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Step 3.2.1

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

$\frac{18 e^{\frac{4}{12}}}{e^{\frac{5}{12} + \frac{4}{12}}} - \frac{12}{e^{\frac{3}{4}}}$

Step 3.2.2

Combine the numerators over the common denominator.

$\frac{18 e^{\frac{4}{12}}}{e^{\frac{5 + 4}{12}}} - \frac{12}{e^{\frac{3}{4}}}$

Step 3.2.3

Add $5$ and $4$ .

$\frac{18 e^{\frac{4}{12}}}{e^{\frac{9}{12}}} - \frac{12}{e^{\frac{3}{4}}}$

Step 3.2.4

Cancel the common factor of $9$ and $12$ .

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Step 3.2.4.1

Factor $3$ out of $9$ .

$\frac{18 e^{\frac{4}{12}}}{e^{\frac{3 (3)}{12}}} - \frac{12}{e^{\frac{3}{4}}}$

Step 3.2.4.2

Cancel the common factors.

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Step 3.2.4.2.1

Factor $3$ out of $12$ .

$\frac{18 e^{\frac{4}{12}}}{e^{\frac{3 \cdot 3}{3 \cdot 4}}} - \frac{12}{e^{\frac{3}{4}}}$

Step 3.2.4.2.2

Cancel the common factor.

$\frac{18 e^{\frac{4}{12}}}{e^{\frac{3 \cdot 3}{3 \cdot 4}}} - \frac{12}{e^{\frac{3}{4}}}$

Step 3.2.4.2.3

Rewrite the expression.

$\frac{18 e^{\frac{4}{12}}}{e^{\frac{3}{4}}} - \frac{12}{e^{\frac{3}{4}}}$

Step 4

Combine the numerators over the common denominator.

$\frac{18 e^{\frac{4}{12}} - 12}{e^{\frac{3}{4}}}$

Step 5

Cancel the common factor of $4$ and $12$ .

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Step 5.1

Factor $4$ out of $4$ .

$\frac{18 e^{\frac{4 (1)}{12}} - 12}{e^{\frac{3}{4}}}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $4$ out of $12$ .

$\frac{18 e^{\frac{4 \cdot 1}{4 \cdot 3}} - 12}{e^{\frac{3}{4}}}$

Step 5.2.2

Cancel the common factor.

$\frac{18 e^{\frac{4 \cdot 1}{4 \cdot 3}} - 12}{e^{\frac{3}{4}}}$

Step 5.2.3

Rewrite the expression.

$\frac{18 e^{\frac{1}{3}} - 12}{e^{\frac{3}{4}}}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{18 e^{\frac{1}{3}} - 12}{e^{\frac{3}{4}}}$

Decimal Form:

$6.19793271 \dots$