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\[{32^{\dfrac{1}{5}}}\] is equal to _____

Last updated date: 29th Feb 2024
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IVSAT 2024
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Hint: Here the given question is the form of exponent. We have to find the equivalent number to the given number \[{32^{\dfrac{1}{5}}}\] . first find the factor or roots of 32 and further simplify using the one of the rules of law of indices \[{\left( {{x^m}} \right)^n} = {x^{mn}}\] we get the required answer which is equivalent to the given number.

Complete step-by-step solution:
Exponential notation is an alternative method of expressing numbers. Exponential numbers take the form an, where a is multiplied by itself n times. In exponential notation, a is termed the base while n is termed the power or exponent or index
Consider the given exponent number
\[ \Rightarrow \,\,{32^{\dfrac{1}{5}}}\]
32 is the 5th root of 2 i.e., 2\[ \times \]2\[ \times \]2\[ \times \]2\[ \times \]2 = 32
Hence, 32 can be written in the form of an exponential number as \[32 = {2^5}\].
\[ \Rightarrow \,{\left( {{2^5}} \right)^{\dfrac{1}{5}}}\]
By applying one of the rule of law of indices If a term with a power is itself raised to a power then the powers are multiplied together. i.e., \[{\left( {{x^m}} \right)^n} = {x^{m \times n}}\]
\[ \Rightarrow \,{2^{\dfrac{5}{5}}}\]
We can cancel the number 5, since it is present in both numerator and the denominator. On simplification, we get
\[ \Rightarrow \,{2^1}\]
As we know the any number to the power 1 which is equal to same number
\[ \Rightarrow \,2\]
Hence, \[{32^{\dfrac{1}{5}}}\] the number which is equal to 2.

Note: When we want to find the fifth root of some number, and if possible we write the number in the form of exponential and then we simplify the number. While determining the roots we have different methods for the different root values. The best way is to factorise the given number then it is easy to solve the number.