Pre-Algebra Examples

$\frac{\frac{\frac{6 x + 7}{x^{2} - 25}}{5 x + 2}}{x^{2} - 25}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{\frac{6 x + 7}{x^{2} - 25}}{5 x + 2} \cdot \frac{1}{x^{2} - 25}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{6 x + 7}{x^{2} - 25} \cdot \frac{1}{5 x + 2} \frac{1}{x^{2} - 25}$

Step 3

Simplify the denominator.

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Step 3.1

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

$\frac{6 x + 7}{x^{2} - 5^{2}} \cdot \frac{1}{5 x + 2} \frac{1}{x^{2} - 25}$

Step 3.2

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

$\frac{6 x + 7}{(x + 5) (x - 5)} \cdot \frac{1}{5 x + 2} \frac{1}{x^{2} - 25}$

Step 4

Multiply $\frac{6 x + 7}{(x + 5) (x - 5)}$ by $\frac{1}{5 x + 2}$ .

$\frac{6 x + 7}{(x + 5) (x - 5) (5 x + 2)} \cdot \frac{1}{x^{2} - 25}$

Step 5

Simplify the denominator.

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Step 5.1

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

$\frac{6 x + 7}{(x + 5) (x - 5) (5 x + 2)} \cdot \frac{1}{x^{2} - 5^{2}}$

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 $a = x$ and $b = 5$ .

$\frac{6 x + 7}{(x + 5) (x - 5) (5 x + 2)} \cdot \frac{1}{(x + 5) (x - 5)}$

Step 6

Multiply $\frac{6 x + 7}{(x + 5) (x - 5) (5 x + 2)} \cdot \frac{1}{(x + 5) (x - 5)}$ .

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Step 6.1

Multiply $\frac{6 x + 7}{(x + 5) (x - 5) (5 x + 2)}$ by $\frac{1}{(x + 5) (x - 5)}$ .

$\frac{6 x + 7}{(x + 5) (x - 5) (5 x + 2) ((x + 5) (x - 5))}$

Step 6.2

Raise $x + 5$ to the power of $1$ .

$\frac{6 x + 7}{{(x + 5)}^{1} (x + 5) (x - 5) (5 x + 2) (x - 5)}$

Step 6.3

Raise $x + 5$ to the power of $1$ .

$\frac{6 x + 7}{{(x + 5)}^{1} {(x + 5)}^{1} (x - 5) (5 x + 2) (x - 5)}$

Step 6.4

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

$\frac{6 x + 7}{{(x + 5)}^{1 + 1} (x - 5) (5 x + 2) (x - 5)}$

Step 6.5

Add $1$ and $1$ .

$\frac{6 x + 7}{{(x + 5)}^{2} (x - 5) (5 x + 2) (x - 5)}$

Step 6.6

Raise $x - 5$ to the power of $1$ .

$\frac{6 x + 7}{{(x + 5)}^{2} ({(x - 5)}^{1} (x - 5)) (5 x + 2)}$

Step 6.7

Raise $x - 5$ to the power of $1$ .

$\frac{6 x + 7}{{(x + 5)}^{2} ({(x - 5)}^{1} {(x - 5)}^{1}) (5 x + 2)}$

Step 6.8

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

$\frac{6 x + 7}{{(x + 5)}^{2} {(x - 5)}^{1 + 1} (5 x + 2)}$

Step 6.9

Add $1$ and $1$ .

$\frac{6 x + 7}{{(x + 5)}^{2} {(x - 5)}^{2} (5 x + 2)}$

Step 7

Split the fraction $\frac{6 x + 7}{{(x + 5)}^{2} {(x - 5)}^{2} (5 x + 2)}$ into two fractions.

$\frac{6 x}{{(x + 5)}^{2} {(x - 5)}^{2} (5 x + 2)} + \frac{7}{{(x + 5)}^{2} {(x - 5)}^{2} (5 x + 2)}$