Question

How do you simplify $\dfrac{{{x}^{\dfrac{5}{3}}}y}{x{{y}^{-\dfrac{1}{2}}}}$ ?

Answer 1

Hint: We have been given exponential functions in fractional format in which variable-x and y have constant positive and negative powers. We have to simplify the given function which has a x-term and a y-term in the numerator as well as the denominator. In order to do this, we shall use certain exponential properties. Thus, we shall add or subtract the powers of the like terms to completely simplify this function.

Complete step by step solution:
Let us suppose a constant, ‘a’ has to be multiplied by itself ‘b’ times. Then we can write it in the exponential form as ${{a}^{b}}$ instead of writing $a\times a\times a\times a\times ......$upto ‘b’ times.
Exponents have their own set of rules and properties according to which they can be manipulated. One of them is that if exponential terms are being multiplied or divided, then their respective powers are added or subtracted respectively.
That is, ${{x}^{a}}.{{x}^{b}}={{x}^{a+b}}$ and $\dfrac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}$ , where a and b are the powers of the base x in the exponential functions.
We have been given the exponent as a function of variable-x and variable-y, $\dfrac{{{x}^{\dfrac{5}{3}}}y}{x{{y}^{-\dfrac{1}{2}}}}$
$\Rightarrow \dfrac{{{x}^{\dfrac{5}{3}}}y}{x{{y}^{-\dfrac{1}{2}}}}=\dfrac{{{x}^{\dfrac{5}{3}}}}{{{x}^{1}}}.\dfrac{{{y}^{1}}}{{{y}^{-\dfrac{1}{2}}}}$
In order to simplify this, we shall subtract the powers of the x-variable terms and also subtract the powers of the y-variable terms.
$\Rightarrow \dfrac{{{x}^{\dfrac{5}{3}}}y}{x{{y}^{-\dfrac{1}{2}}}}={{x}^{\dfrac{5}{3}-1}}.{{y}^{1-\left( -\dfrac{1}{2} \right)}}$
$\Rightarrow \dfrac{{{x}^{\dfrac{5}{3}}}y}{x{{y}^{-\dfrac{1}{2}}}}={{x}^{\dfrac{2}{3}}}.{{y}^{1+\dfrac{1}{2}}}$
$\Rightarrow \dfrac{{{x}^{\dfrac{5}{3}}}y}{x{{y}^{-\dfrac{1}{2}}}}={{x}^{\dfrac{2}{3}}}.{{y}^{\dfrac{3}{2}}}$

Therefore, $\dfrac{{{x}^{\dfrac{5}{3}}}y}{x{{y}^{-\dfrac{1}{2}}}}$ can be simplified as ${{x}^{\dfrac{2}{3}}}.{{y}^{\dfrac{3}{2}}}$.

Note: Some of the other prominent properties of the exponents are used to simplify, understand and solve the complex exponential functions used in mathematical problems. One of them is that the sign of the power of the exponent changes if it taken from the numerator to the denominator or vice versa in a term, that is, ${{x}^{a}}=\dfrac{1}{{{x}^{-a}}}$.