Question

Which expression is equivalent to \[81\]?
(A) \[{2^9}\]
(B) \[{\left( {\dfrac{1}{3}} \right)^{ - 4}}\]
(C) \[{3^{ - 4}}\]
(D) \[{\left( {\dfrac{1}{3}} \right)^4}\]

Answer 1

Hint: In this question, we have given a number and we have to find the expression which is equivalent to this number from the given options. To solve this, we will one by one solve each of the options. The terms involved in options are in exponential form. We will apply the arithmetic operation and on further simplification, we obtain the required solution for the given question.

Complete step by step answer:
To find the expression equivalent to \[81\], we will solve each option one by one. The terms involved in options are in exponential form. We will apply the arithmetic operation to solve.
Consider option (A), we have \[{2^9}\], we can write,
\[ \Rightarrow {2^9} = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\]
\[ \Rightarrow {2^9} = 512\], which is not equal to \[81\].
Now, consider option (B), we have \[{\left( {\dfrac{1}{3}} \right)^{ - 4}}\], we can write,
\[ \Rightarrow {\left( {\dfrac{1}{3}} \right)^{ - 4}} = {\left( 3 \right)^4}\]
\[ \Rightarrow {\left( {\dfrac{1}{3}} \right)^{ - 4}} = 3 \times 3 \times 3 \times 3\]
On simplification, we get
\[ \Rightarrow {\left( {\dfrac{1}{3}} \right)^{ - 4}} = 81\]
Now, consider option (C), we have \[{3^{ - 4}}\], we can write,
\[ \Rightarrow {3^{ - 4}} = \dfrac{1}{{{3^4}}}\]
\[ \Rightarrow {3^{ - 4}} = \dfrac{1}{{3 \times 3 \times 3 \times 3}}\]
On simplification, we get
\[ \Rightarrow {3^{ - 4}} = \dfrac{1}{{81}}\], which is not equal to \[81\].
Now, consider option (D), we have \[{\left( {\dfrac{1}{3}} \right)^4}\], we can write,
\[ \Rightarrow {\left( {\dfrac{1}{3}} \right)^4} = \dfrac{1}{{3 \times 3 \times 3 \times 3}}\]
\[ \Rightarrow {\left( {\dfrac{1}{3}} \right)^4} = \dfrac{1}{{81}}\], which is not equal to \[81\].
Therefore, we get, \[{\left( {\dfrac{1}{3}} \right)^{ - 4}} = 81\]. Hence, option (B) is correct.

Note:
The exponent number is defined as the number of times the number is multiplied by itself. It is represented as \[{a^n}\], where \[a\] is the numeral and \[n\] represent the number of times the number is multiplied. For the exponential numbers we have a law of indices and by applying it we can solve the given options.