Question

Which of the following is not the reciprocal of ${\left( {\dfrac{2}{3}} \right)^4}$?
A. ${\left( {\dfrac{3}{2}} \right)^4}$
B. ${\left( {\dfrac{2}{3}} \right)^{ - 4}}$
C. ${\left( {\dfrac{3}{2}} \right)^{ - 4}}$
D. $\dfrac{{{3^4}}}{{{4^4}}}$

Answer 1

Hint: Here, we have to find which among the above options is not the reciprocal of ${\left( {\dfrac{2}{3}} \right)^4}$. Reciprocal of a fractional number can be defined as turning the number upside down. For example- the reciprocal of $\dfrac{a}{b}$ is $\dfrac{b}{a}$. The product of a number and its reciprocal is always equal to $1$. So, we will check the product of the number and its reciprocal is equal to $1$.

Complete step by step answer:
Reciprocal of a number can be defined as when we divide $1$ by that number. It can also be defined as the number divided by 1 as the number can be written in a fraction with a denominator $1$. For example- the number $2$ can be written as $\dfrac{2}{1}$ and its reciprocal is $\dfrac{1}{2}$. The product of a number and its reciprocal is always equal to $1$, and the reciprocal of a number is also called a multiplicative inverse.

Here, we have to find which among the above options is not the reciprocal of ${\left( {\dfrac{2}{3}} \right)^4}$. Now, we will check the product of the number and the number which is reciprocal of a given fraction is equal to $1$. The reciprocal of $\dfrac{2}{3}$ is $\dfrac{3}{2}$. The reciprocal of ${\left( {\dfrac{2}{3}} \right)^4}$ can be ${\left( {\dfrac{3}{2}} \right)^4}$ as their product equal to $1$. Therefore, option (A) is a reciprocal of ${\left( {\dfrac{2}{3}} \right)^4}$.

The reciprocal of ${\left( {\dfrac{2}{3}} \right)^4}$ can be ${\left( {\dfrac{2}{3}} \right)^{ - 4}}$ as it expresses the given number in a negative exponent. We can write ${\left( {\dfrac{2}{3}} \right)^{ - 4}}$as ${\left( {\dfrac{3}{2}} \right)^4}$ so, their product is also $1$. Therefore, option (B) is a reciprocal of ${\left( {\dfrac{2}{3}} \right)^4}$.

The reciprocal of ${\left( {\dfrac{2}{3}} \right)^4}$ cannot be ${\left( {\dfrac{3}{2}} \right)^{ - 4}}$ as its product does not equal to $1$. We can write ${\left( {\dfrac{3}{2}} \right)^{ - 4}}$as ${\left( {\dfrac{2}{3}} \right)^4}$ and it is not the reciprocal of ${\left( {\dfrac{2}{3}} \right)^4}$. Therefore, option (C) is not a reciprocal of ${\left( {\dfrac{2}{3}} \right)^4}$.

The reciprocal of ${\left( {\dfrac{2}{3}} \right)^4}$ can be $\dfrac{{{3^4}}}{{{4^4}}}$. We can write $\dfrac{{{3^4}}}{{{4^4}}}$ as ${\left( {\dfrac{{{{(3)}^2}}}{{{{(2)}^2}}}} \right)^2} = {\left( {\dfrac{3}{2}} \right)^{ - 4}}$ which is a reciprocal of ${\left( {\dfrac{2}{3}} \right)^4}$ and their product is equal to $1$.

Hence, option (C) is not the reciprocal of ${\left( {\dfrac{2}{3}} \right)^4}$.

Note: Reciprocal of a number can be defined as when we divide $1$ by that number. If we have a number $x$, then its reciprocal will be $\dfrac{1}{x}$ and it can also be written as ${x^{ - 1}}$. The reciprocal of a decimal number is the same as the number when we divide $1$ by that number. For example- the reciprocal of $0.34$ is $\dfrac{1}{{0.34}}$. Note that all the numbers have its reciprocal except $0$ because when we divide something by $0$ it is undefined.