Question

What is the square root of $15$ times the square root of $21$$?$

Answer 1

Hint: To solve the question we need to know the concept of square root. We should know that a number could be written as the product of the square root of the same number. It means $a$ could be written as the product of $\sqrt{a}$, which could be mathematically represented as $a=\sqrt{a}\times \sqrt{a}$ .

Complete step by step solution:
The question asks us to multiply the square root of $15$ to the square root of $21$. We are aware of the fact that $15$ and $21$ are not prime numbers which means both the numbers have factors other than $1$and the number itself. Let us find the factors of the two numbers. The factors of $15$ are:
$\Rightarrow 1,3,5,15$
The factors of $21$ are:
$\Rightarrow 1,3,7,21$
According to the question square root of both the numbers are being multiplied to each other. Mathematically it will be written as:
$\Rightarrow \sqrt{15}\times \sqrt{21}$
On substituting the above numbers with their prime factors we get:
$\Rightarrow \sqrt{3\times 5}\times \sqrt{3\times 7}$
On multiplying the terms we get:
$\Rightarrow \sqrt{3}\times \sqrt{3}\times \sqrt{5}\times \sqrt{7}$
So we can write the product of $\sqrt{3}$ as $3$, on writing it mathematically we get $\sqrt{3}\times \sqrt{3}=3$. On applying this to the above expansion.
$\Rightarrow 3\times \sqrt{5\times 7}$
On further calculation results into:
$\Rightarrow 3\sqrt{35}$
On changing the number in decimal form we get
$\Rightarrow 3\times 5.916$
$\Rightarrow 17.748$

$\therefore $ The square root of $15$times the square root of $21$ is $3\sqrt{35}$ which in decimal form is $17.748$.

Note: Do remember that the square root of a prime number is never a natural number. The back calculation can justify whether the answer is correct or not. For checking the answer we will have to consider the function, where $a$ is an unknown value,
$3\sqrt{35}=a\times \sqrt{15}$
On dividing and calculating the value for $a$ we get:
$\Rightarrow \dfrac{3\sqrt{35}}{\sqrt{15}}$
$\Rightarrow \dfrac{3\sqrt{5\times 7}}{\sqrt{3\times 5}}$
\[\Rightarrow \dfrac{3\sqrt{5}\sqrt{7}}{\sqrt{3}\sqrt{5}}\]
\[\Rightarrow \dfrac{\sqrt{3}\sqrt{3}\sqrt{5}\sqrt{7}}{\sqrt{3}\sqrt{5}}\]
On cancelling the common terms and further calculating the equation we get the value of $a$ as
$\Rightarrow a=\sqrt{3}\sqrt{7}$
$\Rightarrow a=\sqrt{21}$
So the value of $a$ hence found matches with the question, showing the answer to be right.