Question

How do you simplify ${{x}^{\dfrac{5}{8}}}\times {{x}^{\dfrac{3}{8}}}$ ?

Answer 1

Hint: We have been given exponential functions consisting of variable-x have constant positive fractional powers. We have to simplify the given function which has two x-terms getting multiplied. In order to do this, we shall use certain exponential properties. Thus, we shall add 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 like exponential terms are being multiplied, then their respective powers are added.
That is, ${{x}^{a}}.{{x}^{b}}={{x}^{a+b}}$ , 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, ${{x}^{\dfrac{5}{8}}}\times {{x}^{\dfrac{3}{8}}}$. Therefore, adding the powers of these terms, we get
$\Rightarrow {{x}^{\dfrac{5}{8}}}\times {{x}^{\dfrac{3}{8}}}={{x}^{\dfrac{5}{8}+\dfrac{3}{8}}}$
$\Rightarrow {{x}^{\dfrac{5}{8}}}\times {{x}^{\dfrac{3}{8}}}={{x}^{\dfrac{8}{8}}}$
\[\Rightarrow {{x}^{\dfrac{5}{8}}}\times {{x}^{\dfrac{3}{8}}}={{x}^{1}}\]

Therefore, \[{{x}^{\dfrac{5}{8}}}\times {{x}^{\dfrac{3}{8}}}\] can be simplified as $x$.

Note: As we added the powers of the exponential terms which were being multiplied, similarly, the powers of the exponential terms which are being divided are subtracted. That is $\dfrac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}$, where a and b are the powers of the base x in the exponential functions. Likewise, more properties of the exponents are used to simplify, understand and solve the complex exponential functions used in mathematical problems.