Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

Combine $2$ and $\frac{1}{a}$ .

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

Step 1.3

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

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

Step 1.4

To write $\frac{2}{a}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.5

Write each expression with a common denominator of $a \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.1

Multiply $\frac{2}{a}$ by $\frac{2}{2}$ .

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

Step 1.5.2

Move $2$ to the left of $a$ .

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

Step 1.6

Combine the numerators over the common denominator.

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

Step 1.7

Simplify the numerator.

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Step 1.7.1

Multiply $2$ by $2$ .

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

Step 1.7.2

Add $4$ and $1$ .

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

Step 2

Simplify the denominator.

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Step 2.1

Factor $a^{- 2}$ out of $a^{- 1} + 2 a^{- 2}$ .

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Step 2.1.1

Factor $a^{- 2}$ out of $a^{- 1}$ .

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

Step 2.1.2

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

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

Step 2.1.3

Factor $a^{- 2}$ out of $a^{- 2} a + a^{- 2} \cdot 2$ .

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

Step 2.2

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

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Combine fractions.

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Step 4.1

Combine.

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

Step 4.2

Multiply $5$ by $1$ .

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

Step 5

Simplify the denominator.

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Step 5.1

Combine $2$ and $\frac{1}{a^{2}}$ .

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

Step 5.2

Combine $\frac{2}{a^{2}}$ and $a$ .

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

Step 6

Reduce the expression $\frac{2 a}{a^{2}}$ by cancelling the common factors.

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Step 6.1

Factor $a$ out of $2 a$ .

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

Step 6.2

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

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

Step 6.3

Cancel the common factor.

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

Step 6.4

Rewrite the expression.

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

Step 7

Factor $\frac{2}{a}$ out of $\frac{5}{\frac{2}{a} (a + 2)}$ .

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

Step 8

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

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

Step 9

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

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