Basic Math Examples

$y^{- \frac{5}{8}} (y^{\frac{13}{8}} - y^{- \frac{3}{8}})$

Step 1

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

$\frac{1}{y^{\frac{5}{8}}} (y^{\frac{13}{8}} - y^{- \frac{3}{8}})$

Step 2

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

$\frac{1}{y^{\frac{5}{8}}} (y^{\frac{13}{8}} - \frac{1}{y^{\frac{3}{8}}})$

Step 3

Multiply $\frac{1}{y^{\frac{5}{8}}}$ by $y^{\frac{13}{8}} - \frac{1}{y^{\frac{3}{8}}}$ .

$\frac{y^{\frac{13}{8}} - \frac{1}{y^{\frac{3}{8}}}}{y^{\frac{5}{8}}}$

Step 4

Simplify the numerator.

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Step 4.1

To write $y^{\frac{13}{8}}$ as a fraction with a common denominator, multiply by $\frac{y^{\frac{3}{8}}}{y^{\frac{3}{8}}}$ .

$\frac{\frac{y^{\frac{13}{8}} y^{\frac{3}{8}}}{y^{\frac{3}{8}}} - \frac{1}{y^{\frac{3}{8}}}}{y^{\frac{5}{8}}}$

Step 4.2

Combine the numerators over the common denominator.

$\frac{\frac{y^{\frac{13}{8}} y^{\frac{3}{8}} - 1}{y^{\frac{3}{8}}}}{y^{\frac{5}{8}}}$

Step 4.3

Simplify the numerator.

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Step 4.3.1

Multiply $y^{\frac{13}{8}}$ by $y^{\frac{3}{8}}$ by adding the exponents.

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Step 4.3.1.1

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

$\frac{\frac{y^{\frac{13}{8} + \frac{3}{8}} - 1}{y^{\frac{3}{8}}}}{y^{\frac{5}{8}}}$

Step 4.3.1.2

Combine the numerators over the common denominator.

$\frac{\frac{y^{\frac{13 + 3}{8}} - 1}{y^{\frac{3}{8}}}}{y^{\frac{5}{8}}}$

Step 4.3.1.3

Add $13$ and $3$ .

$\frac{\frac{y^{\frac{16}{8}} - 1}{y^{\frac{3}{8}}}}{y^{\frac{5}{8}}}$

Step 4.3.1.4

Divide $16$ by $8$ .

$\frac{\frac{y^{2} - 1}{y^{\frac{3}{8}}}}{y^{\frac{5}{8}}}$

Step 4.3.2

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

$\frac{\frac{y^{2} - 1^{2}}{y^{\frac{3}{8}}}}{y^{\frac{5}{8}}}$

Step 4.3.3

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

$\frac{\frac{(y + 1) (y - 1)}{y^{\frac{3}{8}}}}{y^{\frac{5}{8}}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{(y + 1) (y - 1)}{y^{\frac{3}{8}}} \cdot \frac{1}{y^{\frac{5}{8}}}$

Step 6

Combine.

$\frac{(y + 1) (y - 1) \cdot 1}{y^{\frac{3}{8}} y^{\frac{5}{8}}}$

Step 7

Multiply $y^{\frac{3}{8}}$ by $y^{\frac{5}{8}}$ by adding the exponents.

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Step 7.1

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

$\frac{(y + 1) (y - 1) \cdot 1}{y^{\frac{3}{8} + \frac{5}{8}}}$

Step 7.2

Combine the numerators over the common denominator.

$\frac{(y + 1) (y - 1) \cdot 1}{y^{\frac{3 + 5}{8}}}$

Step 7.3

Add $3$ and $5$ .

$\frac{(y + 1) (y - 1) \cdot 1}{y^{\frac{8}{8}}}$

Step 7.4

Divide $8$ by $8$ .

$\frac{(y + 1) (y - 1) \cdot 1}{y^{1}}$

Step 8

Simplify $y^{1}$ .

$\frac{(y + 1) (y - 1) \cdot 1}{y}$

Step 9

Multiply $(y + 1) (y - 1)$ by $1$ .

$\frac{(y + 1) (y - 1)}{y}$