Question

If f(x)=$x^{\frac{1}{2}},g\left(x\right)=x^{\frac{1}{3}}andh\left(x\right)=x^{\frac{2}{3}}$. Find $\frac{f\left(x\right)+g\left(x\right)}{f\left(x\right)+h\left(x\right)}$

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

(f+g)(x) is f(x)+g(x)

We want to find $\frac{f\left(x\right)+g\left(x\right)}{f\left(x\right)+h\left(x\right)}$

=$\frac{x^{\frac{1}{2}}+x^{\frac{1}{3}}}{x^{\frac{1}{2}}+x^{\frac{2}{3}}}$

We see all the option are in the $x^{\frac{1}{n}}or\frac{1}{x^{\frac{1}{n}}}$.This means we have to find a common factor or simplify the expreesion. For this, we can either guess it by looking at the options or we will replace x with $y^6.y^6$, because we don't want any irrational terms, 6 is the L.C.M of denominator of powers.

$\Rightarrow$ $\frac{x^{\frac{1}{2}}+x^{\frac{1}{3}}}{x^{\frac{1}{2}}+x^{\frac{2}{3}}}$ $=\frac{y^3+y^2}{y^3+y^4}$, where $x=y^{6}ory=x^{\frac{1}{6}}$

=$\frac{(y^3+y^2)}{y^2+y^3\times y}=\frac{1}{y}=$ $\frac{1}{x^{\frac{1}{6}}}$