Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

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

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

Combine the numerators over the common denominator.

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

Step 5.1.3

Add $1$ and $1$ .

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

Step 5.1.4

Divide $2$ by $2$ .

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

Step 5.2

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

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

Step 5.3

Add $1$ and $2$ .

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