Question

Evaluate and simplify: ${\left( {243} \right)^{\dfrac{1}{5}}}$

Answer 1

Hint: Here, we'll use radicals to convert the supplied exponential. To reach the desired answer, we will rewrite the provided number as a product of prime factors and then simplify it using the product rule of exponents. The formula of power rule is \[{\left( {{a^m}} \right)^n} = {a^{mn}}\]

Complete step-by-step solution:
We have given an exponential expression ${\left( {243} \right)^{\dfrac{1}{5}}}$
We will convert the fractional exponent into radical form to simplify. We get
\[ \Rightarrow {\left( {243} \right)^{\dfrac{1}{5}}} = \sqrt[5]{{243}}\]
We will write 243 as a product of prime factors.
We can write 243 as \[243 = 3 \times 3 \times 3 \times 3 \times 3\]
When we rewrite the equation, we get
\[ \Rightarrow {\left( {243} \right)^{\dfrac{1}{5}}} = \sqrt[5]{{3 \times 3 \times 3 \times 3 \times 3}}\]
Now, we'll convert the radical form to the fractional exponent, and we'll get
\[ \Rightarrow {\left( {243} \right)^{\dfrac{1}{5}}} = \sqrt[5]{{{{(3)}^5}}}\]
We know the power rule for exponents as \[{\left( {{a^m}} \right)^n} = {a^{mn}}\]
\[ \Rightarrow {\left( {243} \right)^{\dfrac{1}{5}}} = {\left( {{{\left( 3 \right)}^5}} \right)^{\dfrac{1}{5}}}\]
We can the exponents
\[ \Rightarrow {\left( {243} \right)^{\dfrac{1}{5}}} = 3\]
So, the value of \[{\left( {243} \right)^{\dfrac{1}{5}}}\] is 3.

Note: If the number has a fractional exponent, the fractional exponent's numerator reflects the radical form's power, and the fractional exponent's denominator represents the radical form's index. We can either write the power first, then the index, or the index first, then the power.