Question

Find all the values of \[{\left( { - 32} \right)^{\dfrac{1}{5}}}\].

Answer 1

Hint: Here we will be using the basic law of exponents. i.e. \[{\left( 4 \right)^{\dfrac{1}{2}}} = {\left( 2 \right)^{2 \times \dfrac{1}{2}}} = 2\]. Basic simplification of the powers will be done.
Exponent:Multiplying a number by itself many times gives us an exponent of that number. Exponent of a number is basically some power raised to that number. The power can be natural number, real number or fractional (as the one given in our question\[{\left( { - 32} \right)^{\dfrac{1}{5}}}\]). In case of negative power we take that respective number in denominator, for example,\[{\left( 2 \right)^{ - 2}}\] this can be written as, \[\dfrac{1}{{{{\left( 2 \right)}^2}}}\].

Complete step-by-step answer:
Given that \[{\left( { - 32} \right)^{\dfrac{1}{5}}}\], recall the powers of \[2\], we observe the following list,
\[{2^1} = 2\], \[{2^2} = 4\], \[{2^3} = 8\],\[{2^4} = 16\],\[{2^5} = 32\].
Simplifying the given expression by using multiplicative law of exponent, i.e. \[{\left( {{\text{a}} \times {\text{b}}} \right)^{\text{m}}} = {{\text{a}}^{\text{m}}} \times {{\text{b}}^{\text{m}}}\]. The simplification is given as,
\[{\left( { - 32} \right)^{\dfrac{1}{5}}} = {\left( { - 1} \right)^{\dfrac{1}{5}}} \cdot {\left( {32} \right)^{\dfrac{1}{5}}}\], now, \[\left( { - 1} \right)\]can be written as\[{\left( { - 1} \right)^5}\].
Also, in the question we are given \[32\] which can be written as,\[\left( {32} \right) = {2^5}\] therefore,
\[\ {\left( { - 32} \right)^{\dfrac{1}{5}}} = {\left( { - 1} \right)^{5 \times \dfrac{1}{5}}}{\left( {{2^5}} \right)^{\dfrac{1}{5}}} \\
   = \left( { - 1} \right) \cdot \left( 2 \right) \\
   = - 2 \\
\ \]
Basic multiplication along with some exponent simplification is performed in the above expression.

Therefore, the value of \[{\left( { - 32} \right)^{\dfrac{1}{5}}}\] is \[ - 2\].

Note: In these types of problems in which fractional powers are given, our motive is to write a given number in its exponent form. As we write numbers in exponent form it becomes easy for us to cancel the fractional power ultimately the solution to the problem becomes more simplified. Also, it is important to learn all the exponent properties. Exponent properties help us to convert complex problems into simplified and less complex problems. Some exponent properties are given as, \[{{\text{a}}^{\text{m}}} \times {{\text{a}}^{\text{n}}} = {{\text{a}}^{{\text{m}} + {\text{n}}}}\] here, m and n can be integral, natural, fractional or negative powers. Similarly, another important property of exponent is \[\dfrac{{{{\text{a}}^{\text{m}}}}}{{{{\text{a}}^{\text{n}}}}} = {{\text{a}}^{{\text{m}} - {\text{n}}}}\].