Notes: `(1/x-x/(x^(-1)+1))/(5/x)` Simplify the complex fraction.

Tuesday, 22 July 2014

$\displaystyle\frac{{\frac{{1}}{{x}}-\frac{{x}}{{{{x}}^{{-{1}}}+{1}}}}}{{\frac{{5}}{{x}}}}$ Simplify the complex fraction.

To evaluate the given complex fraction $\displaystyle\frac{{\frac{{1}}{{x}}-\frac{{x}}{{{{x}}^{{-{1}}}+{1}}}}}{{\frac{{5}}{{x}}}}$ , we may simplify first the part $\displaystyle\frac{{x}}{{{{x}}^{{-{1}}}+{1}}}$ .

Apply Law of Exponent: $\displaystyle{{x}}^{{-{n}}}=\frac{{1}}{{{x}}^{{n}}}$ .

Let $\displaystyle{{x}}^{{-{1}}}=\frac{{1}}{{{x}}^{{1}}}$ or $\displaystyle\frac{{1}}{{x}}$ .

$\displaystyle\frac{{x}}{{\frac{{1}}{{x}}+{1}}}$

Let $\displaystyle{1}=\frac{{x}}{{x}}$ to be able to combine similar fractions.

$\displaystyle\frac{{x}}{{\frac{{1}}{{x}}+\frac{{x}}{{x}}}}$

$\displaystyle\frac{{x}}{{\frac{{{1}+{x}}}{{x}}}}$

Flip the fraction at the bottom to proceed to multiplication.

$\displaystyle{x}\cdot\frac{{x}}{{{1}+{x}}}$

$\displaystyle\frac{{{x}}^{{2}}}{{{1}+{x}}}$

Apply $\displaystyle\frac{{x}}{{{{x}}^{{-{1}}}+{1}}}=\frac{{{x}}^{{2}}}{{{1}+{x}}}$ , we get:

$\displaystyle\frac{{\frac{{1}}{{x}}-\frac{{x}}{{{{x}}^{{-{1}}}+{1}}}}}{{\frac{{5}}{{x}}}}$

$\displaystyle\frac{{\frac{{1}}{{x}}-\frac{{{x}}^{{2}}}{{{1}+{x}}}}}{{\frac{{5}}{{x}}}}$

Determine the LCD or least common denominator.

The denominators are $\displaystyle{x}$ and $\displaystyle{\left({1}+{x}\right)}$ . Both are distinct...

Apply Law of Exponent: $\displaystyle{{x}}^{{-{n}}}=\frac{{1}}{{{x}}^{{n}}}$ .

Let $\displaystyle{{x}}^{{-{1}}}=\frac{{1}}{{{x}}^{{1}}}$ or $\displaystyle\frac{{1}}{{x}}$ .

$\displaystyle\frac{{x}}{{\frac{{1}}{{x}}+{1}}}$

Let $\displaystyle{1}=\frac{{x}}{{x}}$ to be able to combine similar fractions.

$\displaystyle\frac{{x}}{{\frac{{1}}{{x}}+\frac{{x}}{{x}}}}$

$\displaystyle\frac{{x}}{{\frac{{{1}+{x}}}{{x}}}}$

Flip the fraction at the bottom to proceed to multiplication.

$\displaystyle{x}\cdot\frac{{x}}{{{1}+{x}}}$

$\displaystyle\frac{{{x}}^{{2}}}{{{1}+{x}}}$

Apply $\displaystyle\frac{{x}}{{{{x}}^{{-{1}}}+{1}}}=\frac{{{x}}^{{2}}}{{{1}+{x}}}$ , we get:

$\displaystyle\frac{{\frac{{1}}{{x}}-\frac{{x}}{{{{x}}^{{-{1}}}+{1}}}}}{{\frac{{5}}{{x}}}}$

$\displaystyle\frac{{\frac{{1}}{{x}}-\frac{{{x}}^{{2}}}{{{1}+{x}}}}}{{\frac{{5}}{{x}}}}$

Determine the LCD or least common denominator.

The denominators are $\displaystyle{x}$ and $\displaystyle{\left({1}+{x}\right)}$ . Both are distinct factors.

Thus, we get the LCD by getting the product of the distinct factors from denominator side of each term.

$\displaystyle{L}{C}{D}={x}\cdot{\left({1}+{x}\right)}{\quad\text{or}\quad}{x}+{{x}}^{{2}}$

Maintain the factored form of the LCD for easier cancellation of common factors on each term.

Multiply each term by the $\displaystyle{L}{C}{D}={x}\cdot{\left({1}+{x}\right)}$ .

$\displaystyle\frac{{\frac{{1}}{{x}}\cdot{x}\cdot{\left({1}+{x}\right)}-\frac{{{x}}^{{2}}}{{{1}+{x}}}\cdot{x}\cdot{\left({1}+{x}\right)}}}{{{\left(\frac{{5}}{{x}}\right)}{x}\cdot{\left({1}+{x}\right)}}}$

Cancel out common factors to get rid of the denominators.

$\displaystyle\frac{{{1}\cdot{\left({1}+{x}\right)}-{{x}}^{{2}}\cdot{x}}}{{{5}\cdot{\left({1}+{x}\right)}}}$

Apply distribution property.

$\displaystyle\frac{{{1}+{x}-{{x}}^{{3}}}}{{{5}+{5}{x}}}$

or $\displaystyle-\frac{{{{x}}^{{3}}-{1}-{x}}}{{{5}{x}+{5}}}$

The complex fraction $\displaystyle\frac{{\frac{{1}}{{x}}-\frac{{x}}{{{{x}}^{{-{1}}}+{1}}}}}{{\frac{{5}}{{x}}}}$ simplifies to $\displaystyle\frac{{{1}+{x}-{{x}}^{{3}}}}{{{5}+{5}{x}}}$