Basic Math Examples

$\frac{\frac{\frac{2 x + 1}{x^{2} - 25}}{4 x^{2} - 1}}{x - 5}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{\frac{2 x + 1}{x^{2} - 25}}{4 x^{2} - 1} \cdot \frac{1}{x - 5}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 x + 1}{x^{2} - 25} \cdot \frac{1}{4 x^{2} - 1} \frac{1}{x - 5}$

Step 3

Simplify the denominator.

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Step 3.1

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

$\frac{2 x + 1}{x^{2} - 5^{2}} \cdot \frac{1}{4 x^{2} - 1} \frac{1}{x - 5}$

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 $b = 5$ .

$\frac{2 x + 1}{(x + 5) (x - 5)} \cdot \frac{1}{4 x^{2} - 1} \frac{1}{x - 5}$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

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

Step 4.2

Rewrite $1$ as $1^{2}$ .

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

Step 4.3

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

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

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $2 x + 1$ .

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Step 5.1.1

Cancel the common factor.

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

Step 5.1.2

Rewrite the expression.

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

Step 5.2

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

Add $1$ and $1$ .

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