Question

The multiplicative inverse of \[{\left( {\dfrac{1}{3}} \right)^{ - 2}}\] is
A. \[9\]
B. \[\dfrac{1}{9}\]
C. \[\dfrac{1}{4}\]
D. None of these

Answer 1

Hint:Multiplicative inverse means the reciprocal of that number. If \[x\] is a number, then its multiplicative inverse is \[\dfrac{1}{x}\] . Here in this question, first we simplify it so as to calculate multiplicative inverse easily. Then, we apply the property that \[{x^{ - 2}}\] is \[\dfrac{1}{{{x^2}}}\] then we will find the square of 3 . Keep in mind the squares. Then we will reciprocate it to find the multiplicative inverse.

Complete step by step answer:
We are given a number that is, \[{\left( {\dfrac{1}{3}} \right)^{ - 2}}\]. First, we will apply the property which states that if we have \[{x^{ - 2}}\] then its value will be equal to \[\dfrac{1}{{{x^2}}}\]. So, we can write \[{\left( {\dfrac{1}{3}} \right)^{ - 2}}\] as \[{\left( {\dfrac{1}{{\dfrac{1}{3}}}} \right)^2}\] which can also be written as \[{3^2}\] and we know that square of \[3\] is \[9\]. so, it will become \[9\]. This implies,
\[{\left( {\dfrac{1}{3}} \right)^{ - 2}} = 9\]

Which means that the multiplicative inverse of \[{\left( {\dfrac{1}{3}} \right)^{ - 2}}\] is equal to multiplicative inverse of \[9\]. And we know that the multiplicative inverse is the reciprocal of that number. If \[x\] is a number, then its multiplicative inverse is \[\dfrac{1}{x}\]. This implies, the multiplicative inverse of \[9\] is \[\dfrac{1}{9}\]. Also both terms are equal so their multiplicative inverse will also be equal. This means that multiplicative inverse of \[{\left( {\dfrac{1}{3}} \right)^{ - 2}}\] is \[\dfrac{1}{9}\].

Hence, option B is the correct answer.

Note:We should keep in mind the definition of multiplicative inverse that the multiplicative inverse of a number is reciprocal of that number. Also we should remember that \[{x^{ - 2}}\] is \[\dfrac{1}{{{x^2}}}\]. Keep in mind the squares of numbers to solve the question easily. Also first always simplify the number so as to find its multiplicative inverse easily. Take care while doing the calculation.