Question

How do you multiply radical 2 times radical 2?

Answer 1

Hint: Given in problem the word radical means root. So the question becomes root 2 times root 2. This word times means the multiplication here. So we have to find the value of that problem. We can solve this problem by directly or by using the laws of indices also. Let’s check both the methods.

Complete step-by-step answer:
Given that multiply radical 2 times radical 2.
Radical two means \[\sqrt 2 \] .
Thus radical 2 times radical 2 \[ = \sqrt 2 \times \sqrt 2 \]
Now multiply the number under root as
 \[ = \sqrt {2 \times 2} \]
 \[ = \sqrt 4 \]
We know that 4 is the square of 2 or square root of 4 is 2.
 \[\sqrt 2 \times \sqrt 2 = 2\]
This is our answer.
So, the correct answer is “2”.

Note: Note that here instead of times if we used the plus or addition sign then the solution changes as
 \[\sqrt 2 + \sqrt 2 = 2\sqrt 2 \] because here we are adding two same quantities. See one more example.
 \[\sqrt 3 + \sqrt 3 = 2\sqrt 3 \] it is not \[\sqrt 3 + \sqrt 3 = 3\sqrt 3 \] . So don’t confuse yourself in product and addition.
Where as \[\sqrt 3 \times \sqrt 3 = \sqrt {3 \times 3} = \sqrt 9 = 3\] . Hope this will help you.
Now talking about indices we can write,
 \[\sqrt 2 = {2^{\dfrac{1}{2}}}\]
So \[\sqrt 2 \times \sqrt 2 = {2^{\dfrac{1}{2}}} \times {2^{\dfrac{1}{2}}}\]
Since the bases are same we can add the power directly,
 \[ \Rightarrow \sqrt 2 \times \sqrt 2 = {2^{\dfrac{1}{2}}} \times {2^{\dfrac{1}{2}}} = {2^{\dfrac{1}{2} + \dfrac{1}{2}}} = {2^1} = 2\]
This is the difference in multiplication and addition of radical quantities.
Given above is a square root. There exist cube roots and many more roots also. They are written as 1 at numerator and the asked root at denominator. Such as the cube root of x is \[{x^{\dfrac{1}{3}}}\] . Fourth root is \[{x^{\dfrac{1}{4}}}\] .