Question

Simplify the given expression ${\left( {{3^4}} \right)^{\dfrac{1}{4}}}$.

Answer 1

Hint: ere, we will find the value for the given exponential expression. We will use the power rule for exponents and simplify the exponent to find the value of the given exponential expression. An exponential expression is an expression with the base and exponents.

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

Complete step-by-step answer:
We are given with an exponential expression ${\left( {{3^4}} \right)^{\dfrac{1}{4}}}$.
Now, we will simplify the exponential expression by using the power rule for exponents ${\left( {{a^m}} \right)^n} = {a^{mn}}$. Therefore, we get
${\left( {{3^4}} \right)^{\dfrac{1}{4}}} = {\left( 3 \right)^{4 \times \dfrac{1}{4}}}$
By multiplying the terms in the exponents, we get
$ \Rightarrow {\left( {{3^4}} \right)^{\dfrac{1}{4}}} = {\left( 3 \right)^1}$
$ \Rightarrow {\left( {{3^4}} \right)^{\dfrac{1}{4}}} = \left( 3 \right)$
Therefore, the simplified value of ${\left( {{3^4}} \right)^{\dfrac{1}{4}}}$ is $3$.

Additional Information:
We will follow the rules for exponents in order to simplify the exponential expression.
First, we will use the zero – exponent rule which says that if any number is raised to the power zero, then it is one i.e., ${a^0} = 1$
We will use the power rule next, which says that if any number is raised to the power and again raised to the power, then the exponents should be multiplied.
We will use the negative exponent rule, which says that the negative exponent in the numerator gets changed to the denominator, then the exponent becomes positive.
We will use the product rule, which says that if we multiply two exponents with the same base then the power has to be added.
We will use the quotient rule, which says that when dividing two exponents with the same base, then the power has to be subtracted.

Note: We know that the power rule states that when an exponent raised to the power is again raised to the power then the power has to be multiplied. If we are given an exponential expression, then the order for exponents rule has to be multiplied in the same order as we know. So, the zero exponent rule, negative exponent rule, product rule and quotient rule is neglected.