Basic Math Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

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

Step 1.3

Combine $- 5$ and $\frac{a}{a}$ .

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{\frac{1 - 5 a}{a}}{5^{2} - a^{- 2}}$

Step 2.2

Rewrite $a^{- 2}$ as ${(a^{- 1})}^{2}$ .

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

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = a^{- 1}$ .

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

Step 2.4

Simplify.

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Step 2.4.1

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

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

Step 2.4.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

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

Step 2.4.3

Combine the numerators over the common denominator.

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

Step 2.4.4

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

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

Step 2.4.5

To write $5$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

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

Step 2.4.6

Combine the numerators over the common denominator.

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

Step 3

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

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

Step 4

Simplify the denominator.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Add $1$ and $1$ .

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

Step 5

Multiply the numerator by the reciprocal of the denominator.

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

Step 6

Cancel the common factor of $a$ .

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Step 6.1

Factor $a$ out of $a^{2}$ .

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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

Step 7

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

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

Step 8

Cancel the common factor of $1 - 5 a$ and $5 a - 1$ .

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Step 8.1

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

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

Step 8.2

Factor $- 1$ out of $- 5 a$ .

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

Step 8.3

Factor $- 1$ out of $- 1 (- 1) - (5 a)$ .

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

Step 8.4

Reorder terms.

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

Step 8.5

Cancel the common factor.

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

Step 8.6

Rewrite the expression.

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

Step 9

Move the negative in front of the fraction.

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