Question

How do you simplify \[{32^{\dfrac{3}{5}}}\] ?

Answer 1

Hint: The given question describes the operation of addition/ subtraction/ multiplication/ division. We need to know how to convert the base term \[32\] into \[{2^n}\] terms. By using the above-mentioned hint we can easily solve the given problem. Also, the question describes the operation of using an algebraic formula to simplify the given term. We need to know how to expand the \[{2^n}\] terms. In this type of question, we would try to eliminate the fraction term by replacing the whole number with small calculations.

Complete step-by-step answer:
The given question is shown below,
 \[{32^{\dfrac{3}{5}}} = ? \to (1)\]


To simplify the given problem we would make changes in the base term.
Here, we have \[32\] as a base term. AS a next step we would write the term \[32\] in the form of \[{2^n}\] .
We know that,
 \[2 \times 2 \times 2 \times 2 \times 2 = 32 = {2^5}\]
So, the equation \[(1)\] becomes,
 \[{32^{\dfrac{3}{5}}} = {\left( {{2^5}} \right)^{\dfrac{3}{5}}} = ? \to (2)\]
We know that,
 \[{\left( {{a^m}} \right)^n} = {\left( a \right)^{m \times n}} \to (3)\]
So, compare the equation \[(2)\] and \[(3)\] we get,
The value of \[m = 5\]
The value of \[n = \dfrac{3}{5}\] and
 \[a\] is \[2\]
Let’s substitute the value of \[m,n\] and \[a\] in the equation \[(3)\] we get,
 \[{\left( {{2^5}} \right)^{\dfrac{3}{5}}} = {\left( 2 \right)^{5 \times \dfrac{3}{5}}}\]
Let’s simplify the equation we get,
 \[{\left( 2 \right)^{5 \times \dfrac{3}{5}}} = {2^3}\]
So we get the answer as \[{2^3}\] . By expanding the term \[{2^3}\] we get,
 \[{2^3} = 2 \times 2 \times 2 = 8\]
So, the final answer is,
 \[{32^{\dfrac{3}{5}}} = 8\]
So, the correct answer is “8”.

Note: In this type of question we would convert the fraction term into a whole number term. Also, note that anything power zero will be one, and zero power anything will be zero. We would try to compare the given question with algebraic conditions. Also, we would use arithmetic operations like addition/ subtraction/ multiplication/ division to solve the given problem.