Algebra Examples

$\frac{2 a + 1}{10 a - 5} \div \frac{10 a}{4 a^{2} - 1}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{2 a + 1}{10 a - 5} \cdot \frac{4 a^{2} - 1}{10 a}$

Step 2

Factor $5$ out of $10 a - 5$ .

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Step 2.1

Factor $5$ out of $10 a$ .

$\frac{2 a + 1}{5 (2 a) - 5} \cdot \frac{4 a^{2} - 1}{10 a}$

Step 2.2

Factor $5$ out of $- 5$ .

$\frac{2 a + 1}{5 (2 a) + 5 (- 1)} \cdot \frac{4 a^{2} - 1}{10 a}$

Step 2.3

Factor $5$ out of $5 (2 a) + 5 (- 1)$ .

$\frac{2 a + 1}{5 (2 a - 1)} \cdot \frac{4 a^{2} - 1}{10 a}$

Step 3

Simplify the numerator.

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Step 3.1

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

$\frac{2 a + 1}{5 (2 a - 1)} \cdot \frac{{(2 a)}^{2} - 1}{10 a}$

Step 3.2

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

$\frac{2 a + 1}{5 (2 a - 1)} \cdot \frac{{(2 a)}^{2} - 1^{2}}{10 a}$

Step 3.3

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

$\frac{2 a + 1}{5 (2 a - 1)} \cdot \frac{(2 a + 1) (2 a - 1)}{10 a}$

Step 4

Cancel the common factor of $2 a - 1$ .

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Step 4.1

Factor $2 a - 1$ out of $5 (2 a - 1)$ .

$\frac{2 a + 1}{(2 a - 1) \cdot 5} \cdot \frac{(2 a + 1) (2 a - 1)}{10 a}$

Step 4.2

Factor $2 a - 1$ out of $(2 a + 1) (2 a - 1)$ .

$\frac{2 a + 1}{(2 a - 1) \cdot 5} \cdot \frac{(2 a - 1) (2 a + 1)}{10 a}$

Step 4.3

Cancel the common factor.

$\frac{2 a + 1}{(2 a - 1) \cdot 5} \cdot \frac{(2 a - 1) (2 a + 1)}{10 a}$

Step 4.4

Rewrite the expression.

$\frac{2 a + 1}{5} \cdot \frac{2 a + 1}{10 a}$

Step 5

Multiply $\frac{2 a + 1}{5}$ by $\frac{2 a + 1}{10 a}$ .

$\frac{(2 a + 1) (2 a + 1)}{5 (10 a)}$

Step 6

Raise $2 a + 1$ to the power of $1$ .

$\frac{{(2 a + 1)}^{1} (2 a + 1)}{5 (10 a)}$

Step 7

Raise $2 a + 1$ to the power of $1$ .

$\frac{{(2 a + 1)}^{1} {(2 a + 1)}^{1}}{5 (10 a)}$

Step 8

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

$\frac{{(2 a + 1)}^{1 + 1}}{5 (10 a)}$

Step 9

Add $1$ and $1$ .

$\frac{{(2 a + 1)}^{2}}{5 (10 a)}$

Step 10

Multiply $10$ by $5$ .

$\frac{{(2 a + 1)}^{2}}{50 a}$

Step 11

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

$\frac{(2 a + 1) (2 a + 1)}{50 a}$

Step 12

Expand $(2 a + 1) (2 a + 1)$ using the FOIL Method.

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Step 12.1

Apply the distributive property.

$\frac{2 a (2 a + 1) + 1 (2 a + 1)}{50 a}$

Step 12.2

Apply the distributive property.

$\frac{2 a (2 a) + 2 a \cdot 1 + 1 (2 a + 1)}{50 a}$

Step 12.3

Apply the distributive property.

$\frac{2 a (2 a) + 2 a \cdot 1 + 1 (2 a) + 1 \cdot 1}{50 a}$

Step 13

Simplify and combine like terms.

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Step 13.1

Simplify each term.

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Step 13.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 \cdot 2 a \cdot a + 2 a \cdot 1 + 1 (2 a) + 1 \cdot 1}{50 a}$

Step 13.1.2

Multiply $a$ by $a$ by adding the exponents.

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Step 13.1.2.1

Move $a$ .

$\frac{2 \cdot 2 (a \cdot a) + 2 a \cdot 1 + 1 (2 a) + 1 \cdot 1}{50 a}$

Step 13.1.2.2

Multiply $a$ by $a$ .

$\frac{2 \cdot 2 a^{2} + 2 a \cdot 1 + 1 (2 a) + 1 \cdot 1}{50 a}$

Step 13.1.3

Multiply $2$ by $2$ .

$\frac{4 a^{2} + 2 a \cdot 1 + 1 (2 a) + 1 \cdot 1}{50 a}$

Step 13.1.4

Multiply $2$ by $1$ .

$\frac{4 a^{2} + 2 a + 1 (2 a) + 1 \cdot 1}{50 a}$

Step 13.1.5

Multiply $2 a$ by $1$ .

$\frac{4 a^{2} + 2 a + 2 a + 1 \cdot 1}{50 a}$

Step 13.1.6

Multiply $1$ by $1$ .

$\frac{4 a^{2} + 2 a + 2 a + 1}{50 a}$

Step 13.2

Add $2 a$ and $2 a$ .

$\frac{4 a^{2} + 4 a + 1}{50 a}$

Step 14

Split the fraction $\frac{4 a^{2} + 4 a + 1}{50 a}$ into two fractions.

$\frac{4 a^{2} + 4 a}{50 a} + \frac{1}{50 a}$

Step 15

Split the fraction $\frac{4 a^{2} + 4 a}{50 a}$ into two fractions.

$\frac{4 a^{2}}{50 a} + \frac{4 a}{50 a} + \frac{1}{50 a}$

Step 16

Cancel the common factor of $4$ and $50$ .

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Step 16.1

Factor $2$ out of $4 a^{2}$ .

$\frac{2 (2 a^{2})}{50 a} + \frac{4 a}{50 a} + \frac{1}{50 a}$

Step 16.2

Cancel the common factors.

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Step 16.2.1

Factor $2$ out of $50 a$ .

$\frac{2 (2 a^{2})}{2 (25 a)} + \frac{4 a}{50 a} + \frac{1}{50 a}$

Step 16.2.2

Cancel the common factor.

$\frac{2 (2 a^{2})}{2 (25 a)} + \frac{4 a}{50 a} + \frac{1}{50 a}$

Step 16.2.3

Rewrite the expression.

$\frac{2 a^{2}}{25 a} + \frac{4 a}{50 a} + \frac{1}{50 a}$

Step 17

Cancel the common factor of $a^{2}$ and $a$ .

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Step 17.1

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

$\frac{a (2 a)}{25 a} + \frac{4 a}{50 a} + \frac{1}{50 a}$

Step 17.2

Cancel the common factors.

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Step 17.2.1

Factor $a$ out of $25 a$ .

$\frac{a (2 a)}{a \cdot 25} + \frac{4 a}{50 a} + \frac{1}{50 a}$

Step 17.2.2

Cancel the common factor.

$\frac{a (2 a)}{a \cdot 25} + \frac{4 a}{50 a} + \frac{1}{50 a}$

Step 17.2.3

Rewrite the expression.

$\frac{2 a}{25} + \frac{4 a}{50 a} + \frac{1}{50 a}$

Step 18

Cancel the common factor of $4$ and $50$ .

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Step 18.1

Factor $2$ out of $4 a$ .

$\frac{2 a}{25} + \frac{2 (2 a)}{50 a} + \frac{1}{50 a}$

Step 18.2

Cancel the common factors.

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Step 18.2.1

Factor $2$ out of $50 a$ .

$\frac{2 a}{25} + \frac{2 (2 a)}{2 (25 a)} + \frac{1}{50 a}$

Step 18.2.2

Cancel the common factor.

$\frac{2 a}{25} + \frac{2 (2 a)}{2 (25 a)} + \frac{1}{50 a}$

Step 18.2.3

Rewrite the expression.

$\frac{2 a}{25} + \frac{2 a}{25 a} + \frac{1}{50 a}$

Step 19

Cancel the common factor of $a$ .

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Step 19.1

Cancel the common factor.

$\frac{2 a}{25} + \frac{2 a}{25 a} + \frac{1}{50 a}$

Step 19.2

Rewrite the expression.

$\frac{2 a}{25} + \frac{2}{25} + \frac{1}{50 a}$