Basic Math Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

Combine $\frac{1}{{(1 + y^{2})}^{\frac{1}{2}}}$ and $y^{2}$ .

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

Step 1.3

To write ${(1 + y^{2})}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(1 + y^{2})}^{\frac{1}{2}}}{{(1 + y^{2})}^{\frac{1}{2}}}$ .

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 1.5

Rewrite $\frac{{(1 + y^{2})}^{\frac{1}{2}} {(1 + y^{2})}^{\frac{1}{2}} - y^{2}}{{(1 + y^{2})}^{\frac{1}{2}}}$ in a factored form.

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Step 1.5.1

Multiply ${(1 + y^{2})}^{\frac{1}{2}}$ by ${(1 + y^{2})}^{\frac{1}{2}}$ by adding the exponents.

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Step 1.5.1.1

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

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

Step 1.5.1.2

Combine the numerators over the common denominator.

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

Step 1.5.1.3

Add $1$ and $1$ .

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

Step 1.5.1.4

Divide $2$ by $2$ .

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

Step 1.5.2

Simplify ${(1 + y^{2})}^{1}$ .

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

Step 1.5.3

Subtract $y^{2}$ from $y^{2}$ .

$\frac{\frac{1 + 0}{{(1 + y^{2})}^{\frac{1}{2}}}}{1 + y^{2}}$

Step 1.5.4

Add $1$ and $0$ .

$\frac{\frac{1}{{(1 + y^{2})}^{\frac{1}{2}}}}{1 + y^{2}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

Multiply $\frac{1}{{(1 + y^{2})}^{\frac{1}{2}}} \cdot \frac{1}{1 + y^{2}}$ .

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Step 3.1

Multiply $\frac{1}{{(1 + y^{2})}^{\frac{1}{2}}}$ by $\frac{1}{1 + y^{2}}$ .

$\frac{1}{{(1 + y^{2})}^{\frac{1}{2}} (1 + y^{2})}$

Step 3.2

Multiply ${(1 + y^{2})}^{\frac{1}{2}}$ by $1 + y^{2}$ by adding the exponents.

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Step 3.2.1

Multiply ${(1 + y^{2})}^{\frac{1}{2}}$ by $1 + y^{2}$ .

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Step 3.2.1.1

Raise $1 + y^{2}$ to the power of $1$ .

$\frac{1}{{(1 + y^{2})}^{\frac{1}{2}} {(1 + y^{2})}^{1}}$

Step 3.2.1.2

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

$\frac{1}{{(1 + y^{2})}^{\frac{1}{2} + 1}}$

Step 3.2.2

Write $1$ as a fraction with a common denominator.

$\frac{1}{{(1 + y^{2})}^{\frac{1}{2} + \frac{2}{2}}}$

Step 3.2.3

Combine the numerators over the common denominator.

$\frac{1}{{(1 + y^{2})}^{\frac{1 + 2}{2}}}$

Step 3.2.4

Add $1$ and $2$ .

$\frac{1}{{(1 + y^{2})}^{\frac{3}{2}}}$