Algebra Examples

$\frac{\frac{5}{a} + 1}{\frac{a}{5} - \frac{5}{a}}$

Step 1

Multiply the numerator and denominator of the fraction by $a \cdot 5$ .

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Step 1.1

Multiply $\frac{\frac{5}{a} + 1}{\frac{a}{5} - \frac{5}{a}}$ by $\frac{a \cdot 5}{a \cdot 5}$ .

$\frac{a \cdot 5}{a \cdot 5} \cdot \frac{\frac{5}{a} + 1}{\frac{a}{5} - \frac{5}{a}}$

Step 1.2

Combine.

$\frac{a \cdot 5 (\frac{5}{a} + 1)}{a \cdot 5 (\frac{a}{5} - \frac{5}{a})}$

Step 2

Apply the distributive property.

$\frac{a \cdot 5 \frac{5}{a} + a \cdot 5 \cdot 1}{a \cdot 5 \frac{a}{5} + a \cdot 5 (- \frac{5}{a})}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $a$ .

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Step 3.1.1

Factor $a$ out of $a \cdot 5$ .

$\frac{a (5) \frac{5}{a} + a \cdot 5 \cdot 1}{a \cdot 5 \frac{a}{5} + a \cdot 5 (- \frac{5}{a})}$

Step 3.1.2

Cancel the common factor.

$\frac{a \cdot 5 \frac{5}{a} + a \cdot 5 \cdot 1}{a \cdot 5 \frac{a}{5} + a \cdot 5 (- \frac{5}{a})}$

Step 3.1.3

Rewrite the expression.

$\frac{5 \cdot 5 + a \cdot 5 \cdot 1}{a \cdot 5 \frac{a}{5} + a \cdot 5 (- \frac{5}{a})}$

Step 3.2

Multiply $5$ by $5$ .

$\frac{25 + a \cdot 5 \cdot 1}{a \cdot 5 \frac{a}{5} + a \cdot 5 (- \frac{5}{a})}$

Step 3.3

Cancel the common factor of $5$ .

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Step 3.3.1

Factor $5$ out of $a \cdot 5$ .

$\frac{25 + a \cdot 5 \cdot 1}{5 \cdot a \frac{a}{5} + a \cdot 5 (- \frac{5}{a})}$

Step 3.3.2

Cancel the common factor.

$\frac{25 + a \cdot 5 \cdot 1}{5 \cdot a \frac{a}{5} + a \cdot 5 (- \frac{5}{a})}$

Step 3.3.3

Rewrite the expression.

$\frac{25 + a \cdot 5 \cdot 1}{a \cdot a + a \cdot 5 (- \frac{5}{a})}$

Step 3.4

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

$\frac{25 + a \cdot 5 \cdot 1}{a^{1} a + a \cdot 5 (- \frac{5}{a})}$

Step 3.5

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

$\frac{25 + a \cdot 5 \cdot 1}{a^{1} a^{1} + a \cdot 5 (- \frac{5}{a})}$

Step 3.6

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

$\frac{25 + a \cdot 5 \cdot 1}{a^{1 + 1} + a \cdot 5 (- \frac{5}{a})}$

Step 3.7

Add $1$ and $1$ .

$\frac{25 + a \cdot 5 \cdot 1}{a^{2} + a \cdot 5 (- \frac{5}{a})}$

Step 3.8

Cancel the common factor of $a$ .

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Step 3.8.1

Move the leading negative in $- \frac{5}{a}$ into the numerator.

$\frac{25 + a \cdot 5 \cdot 1}{a^{2} + a \cdot 5 \frac{- 5}{a}}$

Step 3.8.2

Factor $a$ out of $a \cdot 5$ .

$\frac{25 + a \cdot 5 \cdot 1}{a^{2} + a (5) \frac{- 5}{a}}$

Step 3.8.3

Cancel the common factor.

$\frac{25 + a \cdot 5 \cdot 1}{a^{2} + a \cdot 5 \frac{- 5}{a}}$

Step 3.8.4

Rewrite the expression.

$\frac{25 + a \cdot 5 \cdot 1}{a^{2} + 5 \cdot - 5}$

Step 3.9

Multiply $5$ by $- 5$ .

$\frac{25 + a \cdot 5 \cdot 1}{a^{2} - 25}$

Step 4

Simplify the numerator.

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Step 4.1

Move $5$ to the left of $a$ .

$\frac{25 + 5 \cdot a \cdot 1}{a^{2} - 25}$

Step 4.2

Multiply $5$ by $1$ .

$\frac{25 + 5 a}{a^{2} - 25}$

Step 4.3

Factor $5$ out of $25 + 5 a$ .

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Step 4.3.1

Factor $5$ out of $25$ .

$\frac{5 \cdot 5 + 5 a}{a^{2} - 25}$

Step 4.3.2

Factor $5$ out of $5 \cdot 5 + 5 a$ .

$\frac{5 (5 + a)}{a^{2} - 25}$

Step 5

Simplify the denominator.

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Step 5.1

Rewrite $25$ as $5^{2}$ .

$\frac{5 (5 + a)}{a^{2} - 5^{2}}$

Step 5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = a$ and $b = 5$ .

$\frac{5 (5 + a)}{(a + 5) (a - 5)}$

Step 6

Cancel the common factor of $5 + a$ and $a + 5$ .

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Step 6.1

Reorder terms.

$\frac{5 (a + 5)}{(a + 5) (a - 5)}$

Step 6.2

Cancel the common factor.

$\frac{5 (a + 5)}{(a + 5) (a - 5)}$

Step 6.3

Rewrite the expression.

$\frac{5}{a - 5}$