Question

How do you simplify ${2^{\dfrac{1}{4}}}{.8^{\dfrac{1}{4}}}$?

Answer 1

Hint: We will first write both the products with the same base and then use the fact that: ${x^a}.{x^b} = {x^{a + b}}$. Then, we will get the required answer.

Complete step by step solution:
We are given that we are required to simplify ${2^{\dfrac{1}{4}}}{.8^{\dfrac{1}{4}}}$.
Since we have multiplication of ${2^{\dfrac{1}{4}}}$ and ${8^{\dfrac{1}{4}}}$ here in the given expression with us.
Now, we have the bases of the products as 2 and 8 respectively and we also know that the cube of 8 is 2.
So, we can write this statement as follows:-
$ \Rightarrow 8 = 2 \times 2 \times 2$
This can also be written as follows:-
$ \Rightarrow 8 = {2^3}$
Taking the power of $\dfrac{1}{4}$ on both the sides of the sides of the above equation, we will then obtain the following equation with us:-
$ \Rightarrow {8^{\dfrac{1}{4}}} = {\left( {{2^3}} \right)^{\dfrac{1}{4}}}$
Simplifying the right hand side of the above equation by combining both the powers, we will then obtain the following equation with us:-
$ \Rightarrow {8^{\dfrac{1}{4}}} = {2^{\dfrac{3}{4}}}$
Putting this in the given expression which is required to be simplified, we will then obtain the following equation with us:-
$ \Rightarrow {2^{\dfrac{1}{4}}}{.8^{\dfrac{1}{4}}} = {2^{\dfrac{1}{4}}}{.2^{\dfrac{3}{4}}}$
Simplifying the right hand side of the above mentioned equation by combining both the terms in product, we will then obtain the following equation with us:-
$ \Rightarrow {2^{\dfrac{1}{4}}}{.8^{\dfrac{1}{4}}} = {2^{\dfrac{1}{4} + \dfrac{3}{4}}}$
Simplifying the right hand side of the above mentioned equation further, we will then obtain the following equation with us:-
$ \Rightarrow {2^{\dfrac{1}{4}}}{.8^{\dfrac{1}{4}}} = 2$

Thus, we have the required answer as 2.

Note: The students must note that we have used the fact given by the following expression: ${\left( {{x^a}} \right)^b} = {x^{ab}}$ while simplifying ${8^{\dfrac{1}{4}}} = {\left( {{2^3}} \right)^{\dfrac{1}{4}}}$ to get ${8^{\dfrac{1}{4}}} = {2^{\dfrac{3}{4}}}$ . We have just replaced $x$ by 2, a by 3 and b by $\dfrac{1}{4}$ in ${\left( {{x^a}} \right)^b} = {x^{ab}}$ .
The students must also note that we have used the fact given by the following expression: ${x^a}.{x^b} = {x^{a + b}}$ while simplifying ${2^{\dfrac{1}{4}}}{.8^{\dfrac{1}{4}}} = {2^{\dfrac{1}{4}}}{.2^{\dfrac{3}{4}}}$ to get ${2^{\dfrac{1}{4}}}{.8^{\dfrac{1}{4}}} = {2^{\dfrac{1}{4} + \dfrac{3}{4}}}$ . We have just replaced $x$ by 2, a by $\dfrac{1}{4}$ and b by $\dfrac{3}{4}$ in ${x^a}.{x^b} = {x^{a + b}}$ .