Question

How do you evaluate and simplify \[{\left( { - 243} \right)^{\dfrac{1}{5}}}\]?

Answer 1

Hint: Here, we will convert the given exponential in the form of radicals. We will rewrite the given number as a product of prime factors and then simplify it using the product rule of exponents to get the required answer.

Formula Used:
Power rule for Exponents: \[{\left( {{a^m}} \right)^n} = {a^{mn}}\]

Complete Step by Step Solution:
We are given an exponential expression \[{\left( { - 243} \right)^{\dfrac{1}{5}}}\].
Now, we will convert the fractional exponent into the radical form to simplify the given number. So, we get
\[ \Rightarrow {\left( { - 243} \right)^{\dfrac{1}{5}}} = \sqrt[5]{{ - 243}}\]
Now, we will rewrite 243 as the product of prime factors.
So, we can write the number 243 as \[243 = 3 \times 3 \times 3 \times 3 \times 3\].
Since the root is odd, the number should be negative. So, we get
 \[ \Rightarrow {\left( { - 243} \right)^{\dfrac{1}{5}}} = \sqrt[5]{{ - 3 \times - 3 \times - 3 \times - 3 \times - 3}}\]
Now, by rewriting the equation, we get
\[ \Rightarrow {\left( { - 243} \right)^{\dfrac{1}{5}}} = \sqrt[5]{{{{\left( { - 3} \right)}^5}}}\]
Now, we will convert the radical form again into the fractional exponent, we get
\[ \Rightarrow {\left( { - 243} \right)^{\dfrac{1}{5}}} = {\left( {{{\left( { - 3} \right)}^5}} \right)^{\dfrac{1}{5}}}\]
Power rule for Exponents: \[{\left( {{a^m}} \right)^n} = {a^{mn}}\]
Now, by using the power rule for exponents, we get
\[ \Rightarrow {\left( { - 243} \right)^{\dfrac{1}{5}}} = \left( {{{\left( { - 3} \right)}^{5 \times \dfrac{1}{5}}}} \right)\]
Now, by canceling the exponents, we get
\[ \Rightarrow {\left( { - 243} \right)^{\dfrac{1}{5}}} = \left( { - 3} \right)\].

Therefore, the value of \[{\left( { - 243} \right)^{\dfrac{1}{5}}}\] is \[\left( { - 3} \right)\].

Note:
If the number has a fractional exponent, then the numerator of a fractional exponent represents the power of the radical form and the denominator of a fractional exponent represents the index of the radical form. We can write power at first and then the index or we can write index at first and then the power. This can be written in either way to express the number in the Radical form. In simple, we write the fractional exponent of a number \[{x^{\dfrac{p}{q}}}\] in the Radical form as \[\sqrt[q]{{{x^p}}}\] or \[{\left( {\sqrt[q]{x}} \right)^p}\]. The radical form can be written only with the Radical sign.