Precalculus Examples

$\frac{{(6 - x^{2})}^{\frac{1}{2}} + x^{2} {(6 - x^{2})}^{- \frac{1}{2}}}{6 - x^{2}}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{{(6 - x^{2})}^{\frac{1}{2}} + x^{2} \frac{1}{{(6 - x^{2})}^{\frac{1}{2}}}}{6 - x^{2}}$

Step 1.2

Combine $x^{2}$ and $\frac{1}{{(6 - x^{2})}^{\frac{1}{2}}}$ .

$\frac{{(6 - x^{2})}^{\frac{1}{2}} + \frac{x^{2}}{{(6 - x^{2})}^{\frac{1}{2}}}}{6 - x^{2}}$

Step 1.3

To write ${(6 - x^{2})}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(6 - x^{2})}^{\frac{1}{2}}}{{(6 - x^{2})}^{\frac{1}{2}}}$ .

$\frac{\frac{{(6 - x^{2})}^{\frac{1}{2}} {(6 - x^{2})}^{\frac{1}{2}}}{{(6 - x^{2})}^{\frac{1}{2}}} + \frac{x^{2}}{{(6 - x^{2})}^{\frac{1}{2}}}}{6 - x^{2}}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{\frac{{(6 - x^{2})}^{\frac{1}{2}} {(6 - x^{2})}^{\frac{1}{2}} + x^{2}}{{(6 - x^{2})}^{\frac{1}{2}}}}{6 - x^{2}}$

Step 1.5

Reorder terms.

$\frac{\frac{x^{2} + {(6 - x^{2})}^{\frac{1}{2}} {(6 - x^{2})}^{\frac{1}{2}}}{{(- x^{2} + 6)}^{\frac{1}{2}}}}{6 - x^{2}}$

Step 1.6

Rewrite $\frac{x^{2} + {(6 - x^{2})}^{\frac{1}{2}} {(6 - x^{2})}^{\frac{1}{2}}}{{(- x^{2} + 6)}^{\frac{1}{2}}}$ in a factored form.

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Step 1.6.1

Multiply ${(6 - x^{2})}^{\frac{1}{2}}$ by ${(6 - x^{2})}^{\frac{1}{2}}$ by adding the exponents.

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Step 1.6.1.1

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

$\frac{\frac{x^{2} + {(6 - x^{2})}^{\frac{1}{2} + \frac{1}{2}}}{{(- x^{2} + 6)}^{\frac{1}{2}}}}{6 - x^{2}}$

Step 1.6.1.2

Combine the numerators over the common denominator.

$\frac{\frac{x^{2} + {(6 - x^{2})}^{\frac{1 + 1}{2}}}{{(- x^{2} + 6)}^{\frac{1}{2}}}}{6 - x^{2}}$

Step 1.6.1.3

Add $1$ and $1$ .

$\frac{\frac{x^{2} + {(6 - x^{2})}^{\frac{2}{2}}}{{(- x^{2} + 6)}^{\frac{1}{2}}}}{6 - x^{2}}$

Step 1.6.1.4

Divide $2$ by $2$ .

$\frac{\frac{x^{2} + {(6 - x^{2})}^{1}}{{(- x^{2} + 6)}^{\frac{1}{2}}}}{6 - x^{2}}$

Step 1.6.2

Simplify ${(6 - x^{2})}^{1}$ .

$\frac{\frac{x^{2} + 6 - x^{2}}{{(- x^{2} + 6)}^{\frac{1}{2}}}}{6 - x^{2}}$

Step 1.6.3

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

$\frac{\frac{0 + 6}{{(- x^{2} + 6)}^{\frac{1}{2}}}}{6 - x^{2}}$

Step 1.6.4

Add $0$ and $6$ .

$\frac{\frac{6}{{(- x^{2} + 6)}^{\frac{1}{2}}}}{6 - x^{2}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{6}{{(- x^{2} + 6)}^{\frac{1}{2}}} \cdot \frac{1}{6 - x^{2}}$

Step 3

Multiply $\frac{6}{{(- x^{2} + 6)}^{\frac{1}{2}}} \cdot \frac{1}{6 - x^{2}}$ .

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Step 3.1

Multiply $\frac{6}{{(- x^{2} + 6)}^{\frac{1}{2}}}$ by $\frac{1}{6 - x^{2}}$ .

$\frac{6}{{(- x^{2} + 6)}^{\frac{1}{2}} (6 - x^{2})}$

Step 3.2

Reorder terms.

$\frac{6}{{(- x^{2} + 6)}^{\frac{1}{2}} (- x^{2} + 6)}$

Step 3.3

Multiply ${(- x^{2} + 6)}^{\frac{1}{2}}$ by $- x^{2} + 6$ by adding the exponents.

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Step 3.3.1

Multiply ${(- x^{2} + 6)}^{\frac{1}{2}}$ by $- x^{2} + 6$ .

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Step 3.3.1.1

Raise $- x^{2} + 6$ to the power of $1$ .

$\frac{6}{{(- x^{2} + 6)}^{\frac{1}{2}} {(- x^{2} + 6)}^{1}}$

Step 3.3.1.2

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

$\frac{6}{{(- x^{2} + 6)}^{\frac{1}{2} + 1}}$

Step 3.3.2

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

$\frac{6}{{(- x^{2} + 6)}^{\frac{1}{2} + \frac{2}{2}}}$

Step 3.3.3

Combine the numerators over the common denominator.

$\frac{6}{{(- x^{2} + 6)}^{\frac{1 + 2}{2}}}$

Step 3.3.4

Add $1$ and $2$ .

$\frac{6}{{(- x^{2} + 6)}^{\frac{3}{2}}}$