Question

Simplify the given expression: \[{3^{\dfrac{1}{3}}} \times {3^{\dfrac{1}{4}}}\].

Answer 1

Hint: Here, we will use the rule of exponents for products of two terms with same base and different exponents. Then, we will use addition of fractions to simplify the expression and get the required answer.

Formula used: We will use the rule of exponents \[{a^m} \times {a^n} = {a^{m + n}}\].

Complete step-by-step answer:
Exponents are the powers to which a number is raised. It denotes the number of times a number is multiplied by itself.
For example: \[{2^{30}}\] means that 2 is multiplied by itself 30 times. Here, 30 is the exponent/power and 2 is the base.
We will use the rule of exponents and addition of fractions to simplify the given expression.
The given expression is a product of two terms with equal bases 3 and different exponents.
We know that by the rule of exponents \[{a^m} \times {a^n} = {a^{m + n}}\].
Substituting \[a = 3\], \[m = \dfrac{1}{3}\], and \[n = \dfrac{1}{4}\] in the rule, we get
\[ \Rightarrow {3^{\dfrac{1}{3}}} \times {3^{\dfrac{1}{4}}} = {3^{\dfrac{1}{3} + \dfrac{1}{4}}}\]
Now, we will simplify the expression on the right hand side.
The L.C.M. of the denominators 3 and 4 is 12.
Rewriting the fractions \[\dfrac{1}{3}\] and \[\dfrac{1}{4}\] with the denominator 12, we get
\[ \Rightarrow {3^{\dfrac{1}{3}}} \times {3^{\dfrac{1}{4}}} = {3^{\dfrac{4}{{12}} + \dfrac{3}{{12}}}}\]
Adding the two fractions, we get
\[ \Rightarrow {3^{\dfrac{1}{3}}} \times {3^{\dfrac{1}{4}}} = {3^{\dfrac{7}{{12}}}}\]
\[\therefore\] We get the value of the expression \[{3^{\dfrac{1}{3}}} \times {3^{\dfrac{1}{4}}}\] as \[{3^{\dfrac{7}{{12}}}}\].

Note: For solving this question, we need to know the rules of exponents. It states that if two terms with the same base and different exponents are multiplied, then the result is equal to the base raised to the sum of the different exponents. A common mistake we can make is to use the rule of exponent as \[{a^m} \times {a^n} = {a^{m - n}}\], and obtain the answer \[{3^{\dfrac{1}{{12}}}}\] which is incorrect.