Algebra Examples

$\frac{e^{8}}{e^{- 4} + e^{- 7}}$

Step 1

Simplify the denominator.

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Step 1.1

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

$\frac{e^{8}}{\frac{1}{e^{4}} + e^{- 7}}$

Step 1.2

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

$\frac{e^{8}}{\frac{1}{e^{4}} + \frac{1}{e^{7}}}$

Step 1.3

To write $\frac{1}{e^{4}}$ as a fraction with a common denominator, multiply by $\frac{e^{3}}{e^{3}}$ .

$\frac{e^{8}}{\frac{1}{e^{4}} \cdot \frac{e^{3}}{e^{3}} + \frac{1}{e^{7}}}$

Step 1.4

Write each expression with a common denominator of $e^{7}$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

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

$\frac{e^{8}}{\frac{e^{3}}{e^{4} e^{3}} + \frac{1}{e^{7}}}$

Step 1.4.2

Multiply $e^{4}$ by $e^{3}$ by adding the exponents.

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Step 1.4.2.1

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

$\frac{e^{8}}{\frac{e^{3}}{e^{4 + 3}} + \frac{1}{e^{7}}}$

Step 1.4.2.2

Add $4$ and $3$ .

$\frac{e^{8}}{\frac{e^{3}}{e^{7}} + \frac{1}{e^{7}}}$

Step 1.5

Combine the numerators over the common denominator.

$\frac{e^{8}}{\frac{e^{3} + 1}{e^{7}}}$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Rewrite $1$ as $1^{3}$ .

$\frac{e^{8}}{\frac{e^{3} + 1^{3}}{e^{7}}}$

Step 1.6.2

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = e$ and $b = 1$ .

$\frac{e^{8}}{\frac{(e + 1) (e^{2} - e \cdot 1 + 1^{2})}{e^{7}}}$

Step 1.6.3

Simplify.

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Step 1.6.3.1

Multiply $- 1$ by $1$ .

$\frac{e^{8}}{\frac{(e + 1) (e^{2} - e + 1^{2})}{e^{7}}}$

Step 1.6.3.2

One to any power is one.

$\frac{e^{8}}{\frac{(e + 1) (e^{2} - e + 1)}{e^{7}}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$e^{8} \frac{e^{7}}{(e + 1) (e^{2} - e + 1)}$

Step 3

Multiply $e^{8} \frac{e^{7}}{(e + 1) (e^{2} - e + 1)}$ .

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Step 3.1

Combine $e^{8}$ and $\frac{e^{7}}{(e + 1) (e^{2} - e + 1)}$ .

$\frac{e^{8} e^{7}}{(e + 1) (e^{2} - e + 1)}$

Step 3.2

Multiply $e^{8}$ by $e^{7}$ 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.

$\frac{e^{8 + 7}}{(e + 1) (e^{2} - e + 1)}$

Step 3.2.2

Add $8$ and $7$ .

$\frac{e^{15}}{(e + 1) (e^{2} - e + 1)}$