Question

How do you simplify $\dfrac{3{{x}^{2}}{{y}^{2}}}{2{{x}^{-1}}.4y{{x}^{2}}}$ and write it using only positive exponents?

Answer 1

Hint: In this question, we have to simplify the given algebraic expression. Thus, we will use the exponent rule and the basic mathematical rules to get the required solution for the problem. As we know, an exponent is a quantity that represents the power, where the given number or expression is to be raised. Thus, we start solving this problem by applying the exponent formula ${{a}^{-b}}={{\left( \dfrac{1}{a} \right)}^{b}}$ on the denominator in the given algebraic expression. Then, we will divide the numerator and the denominator, to get the required result for the problem.

Complete step by step solution:
According to the question, we have to simplify an algebraic expression.
Thus, we will use the exponent formula and the basic mathematical rules to get the required solution.
The algebraic expression given to us is $\dfrac{3{{x}^{2}}{{y}^{2}}}{2{{x}^{-1}}.4y{{x}^{2}}}$ -------- (1)
Now, we will apply the exponent formula, which states that if we have a negative exponent, then we have to write the reciprocal of the given number with a positive exponent, that is ${{a}^{-b}}={{\left( \dfrac{1}{a} \right)}^{b}}$ . Thus, we will first express the negative exponent 1 as a positive exponent using the same formula, we get
$\Rightarrow {{x}^{-1}}={{\left( \dfrac{1}{x} \right)}^{1}}$ --------- (2)
Thus, we will substitute the value of equation (2) in equation (1), we get
$\Rightarrow \dfrac{3{{x}^{2}}{{y}^{2}}}{2\dfrac{1}{x}.4y{{x}^{2}}}$
Now, we will shift the denominator of denominator x in the numerator, we get
$\Rightarrow \dfrac{3{{x}^{2}}{{y}^{2}}.x}{2.4y{{x}^{2}}}$
Now, on further simplification of the above equation, we get
$\Rightarrow \dfrac{3{{x}^{3}}{{y}^{2}}}{8y{{x}^{2}}}$
Now, we see that in the denominator and the numerator, the variable x and y are common, therefore, we will cancel out them, to get more simplified expression, we get
$\Rightarrow \dfrac{3xy}{8}$ which is the required answer.
Therefore, for the algebraic expression $\dfrac{3{{x}^{2}}{{y}^{2}}}{2{{x}^{-1}}.4y{{x}^{2}}}$, its simplified value is equal to $\dfrac{3xy}{8}$ .

Note:
While solving this problem, always mention the formula you are using to avoid mathematical errors. Do not further solve the answer, because both the x and y variables are different from each other.