Question

How do you simplify $3{{x}^{-2}}{{y}^{-5}}$ 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 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. It is expressed in the form of ${{a}^{b}}$ , where b is the exponent which is any real number and a is the number or any expression which is to be raised. Thus, we start solving this problem by applying the exponent formula ${{a}^{-b}}={{\left( \dfrac{1}{a} \right)}^{b}}$ in the gives algebraic expression, to get an accurate 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 to get the required solution.
The algebraic expression given to us is $3{{x}^{-2}}{{y}^{-5}}$ -------- (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 2 as a positive exponent using the same formula, we get
$\Rightarrow {{x}^{-2}}={{\left( \dfrac{1}{x} \right)}^{2}}$ --------- (2)
Now, we will make the negative exponent 5 using the exponent formula ${{a}^{-b}}={{\left( \dfrac{1}{a} \right)}^{b}}$, we get
$\Rightarrow {{y}^{-5}}={{\left( \dfrac{1}{y} \right)}^{5}}$ ---------- (3)
Thus, we will substitute the value of equation (2) and (3) in equation (1), we get
$\Rightarrow 3{{\left( \dfrac{1}{x} \right)}^{2}}{{\left( \dfrac{1}{y} \right)}^{5}}$
On further simplification, we get
\[\Rightarrow \dfrac{3}{{{x}^{2}}{{y}^{5}}}\] which is our required solution.
Therefore, for the algebraic expression $3{{x}^{-2}}{{y}^{-5}}$, its simplified value is equal to \[\dfrac{3}{{{x}^{2}}{{y}^{5}}}\] .

Note:
Always mention all the formulas you are using to avoid confusion and mathematical errors. While solving this problem, do to further solve the answer, because both the x and y variables are different from each other.