Basic Math Examples

$\frac{5 z^{- 1} + 3}{25 z^{- 2} - 9}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{5 \frac{1}{z} + 3}{25 z^{- 2} - 9}$

Step 1.2

Combine $5$ and $\frac{1}{z}$ .

$\frac{\frac{5}{z} + 3}{25 z^{- 2} - 9}$

Step 1.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{z}{z}$ .

$\frac{\frac{5}{z} + \frac{3 z}{z}}{25 z^{- 2} - 9}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{\frac{5 + 3 z}{z}}{25 z^{- 2} - 9}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $25 z^{- 2}$ as ${(5 z^{- 1})}^{2}$ .

$\frac{\frac{5 + 3 z}{z}}{{(5 z^{- 1})}^{2} - 9}$

Step 2.2

Rewrite $9$ as $3^{2}$ .

$\frac{\frac{5 + 3 z}{z}}{{(5 z^{- 1})}^{2} - 3^{2}}$

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 $a = 5 z^{- 1}$ and $b = 3$ .

$\frac{\frac{5 + 3 z}{z}}{(5 z^{- 1} + 3) (5 z^{- 1} - 3)}$

Step 2.4

Simplify.

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Step 2.4.1

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

$\frac{\frac{5 + 3 z}{z}}{(5 \frac{1}{z} + 3) (5 z^{- 1} - 3)}$

Step 2.4.2

Combine $5$ and $\frac{1}{z}$ .

$\frac{\frac{5 + 3 z}{z}}{(\frac{5}{z} + 3) (5 z^{- 1} - 3)}$

Step 2.4.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{z}{z}$ .

$\frac{\frac{5 + 3 z}{z}}{(\frac{5}{z} + \frac{3 z}{z}) (5 z^{- 1} - 3)}$

Step 2.4.4

Combine the numerators over the common denominator.

$\frac{\frac{5 + 3 z}{z}}{\frac{5 + 3 z}{z} (5 z^{- 1} - 3)}$

Step 2.4.5

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

$\frac{\frac{5 + 3 z}{z}}{\frac{5 + 3 z}{z} (5 \frac{1}{z} - 3)}$

Step 2.4.6

Combine $5$ and $\frac{1}{z}$ .

$\frac{\frac{5 + 3 z}{z}}{\frac{5 + 3 z}{z} (\frac{5}{z} - 3)}$

Step 2.4.7

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{z}{z}$ .

$\frac{\frac{5 + 3 z}{z}}{\frac{5 + 3 z}{z} (\frac{5}{z} - 3 \cdot \frac{z}{z})}$

Step 2.4.8

Combine $- 3$ and $\frac{z}{z}$ .

$\frac{\frac{5 + 3 z}{z}}{\frac{5 + 3 z}{z} (\frac{5}{z} + \frac{- 3 z}{z})}$

Step 2.4.9

Combine the numerators over the common denominator.

$\frac{\frac{5 + 3 z}{z}}{\frac{5 + 3 z}{z} \cdot \frac{5 - 3 z}{z}}$

Step 3

Multiply $\frac{5 + 3 z}{z}$ by $\frac{5 - 3 z}{z}$ .

$\frac{\frac{5 + 3 z}{z}}{\frac{(5 + 3 z) (5 - 3 z)}{z \cdot z}}$

Step 4

Simplify the denominator.

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Step 4.1

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

$\frac{\frac{5 + 3 z}{z}}{\frac{(5 + 3 z) (5 - 3 z)}{z^{1} z}}$

Step 4.2

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

$\frac{\frac{5 + 3 z}{z}}{\frac{(5 + 3 z) (5 - 3 z)}{z^{1} z^{1}}}$

Step 4.3

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

$\frac{\frac{5 + 3 z}{z}}{\frac{(5 + 3 z) (5 - 3 z)}{z^{1 + 1}}}$

Step 4.4

Add $1$ and $1$ .

$\frac{\frac{5 + 3 z}{z}}{\frac{(5 + 3 z) (5 - 3 z)}{z^{2}}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{5 + 3 z}{z} \cdot \frac{z^{2}}{(5 + 3 z) (5 - 3 z)}$

Step 6

Cancel the common factor of $5 + 3 z$ .

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Step 6.1

Cancel the common factor.

$\frac{5 + 3 z}{z} \cdot \frac{z^{2}}{(5 + 3 z) (5 - 3 z)}$

Step 6.2

Rewrite the expression.

$\frac{1}{z} \cdot \frac{z^{2}}{5 - 3 z}$

Step 7

Cancel the common factor of $z$ .

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Step 7.1

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

$\frac{1}{z} \cdot \frac{z \cdot z}{5 - 3 z}$

Step 7.2

Cancel the common factor.

$\frac{1}{z} \cdot \frac{z \cdot z}{5 - 3 z}$

Step 7.3

Rewrite the expression.

$\frac{z}{5 - 3 z}$