Question

How do you simplify $\sqrt 5 \sqrt {15} $?

Answer 1

Hint: We will first find the prime factorisation of both 5 and 15. Then we will make use of the fact that $\sqrt a \sqrt b = \sqrt {a \times b} $. Then, we will take out the factors which are common in both the numbers and thus we have the answer.

Complete step by step solution:
We are given that we are required to simplify $\sqrt 5 \sqrt {15} $.
We have two numbers involved here, 5 and 15.
We know that 5 is itself a prime number, therefore, it cannot be broken into further factors.
And, we also know that 15 is divisible by both 5 and 3. Therefore, 15 can be written as multiple of 3 and 5.
Therefore, we can write it as follows:-
$ \Rightarrow 15 = 3 \times 5$
Now, we know that we have the following expression as a fact:-
For any non – negative real numbers a and b, we have: $\sqrt a \sqrt b = \sqrt {a \times b} $.
Replacing a by 5 and b by 15, we will get the following expression with us:-
$ \Rightarrow \sqrt 5 \sqrt {15} = \sqrt {5 \times 15} $
Since we know that 15 is divisible by both 5 and 3. Therefore, 15 can be written as multiple of 3 and 5.
Therefore, we can write the above mentioned expression as follows:-
$ \Rightarrow \sqrt 5 \sqrt {15} = \sqrt {5 \times 3 \times 5} $
We observe that 5 is twice inside the square root given above on the right hand side, therefore, we can take 1 5 out of the square root and get the following expression:-
$ \Rightarrow \sqrt 5 \sqrt {15} = 5\sqrt 3 $
Thus, we have the required answer.

Note: The students must commit to the memory the following formula:-
For any non – negative real numbers a and b, we have: $\sqrt a \sqrt b = \sqrt {a \times b} $.
We can also prove it as follows:-
If we have two negative real numbers a and b, then we can write $\sqrt a \sqrt b $ as ${a^{\dfrac{1}{2}}}{b^{\dfrac{1}{2}}}$.