Question

Solve the expression: $\sqrt {98} \div \sqrt 2 $
(A) $9$
(B) $5$
(C) $7$
(D) $3$

Answer 1

Hint: Square root of a number is a value, which when multiplied by itself gives the original number. Suppose, ‘x’ is the square root of ‘y’, then it is represented as $x = \sqrt y $ or we can express the same equation as ${x^2} = y$ . Here we can see that both the numbers $98$ and $2$ are not perfect squares. Now, to simplify the square roots and find the value of required expression, we first do the prime factorization of the numbers and take the factors occurring in pairs outside of the square root radical.

Complete step-by-step solution:
Given expression is $\sqrt {98} \div \sqrt 2 $.
We first represent the division of the two square roots in the fraction form. So, we get,
\[ \Rightarrow \dfrac{{\sqrt {98} }}{{\sqrt 2 }}\]
$98$ can be factorized as,
$98 = 2 \times 7 \times 7$
Now, expressing the prime factorization in powers and exponents, we get,
$98 = 2 \times {7^2}$
We can see that $2$ is multiplied once and hence the power of $2$ is one. Similarly, $7$ is multiplied twice. So, we have the power of $7$ as two.
Now, \[\sqrt {98} = \sqrt {2 \times {7^2}} \]
Since we know that ${7^2}$ is a perfect square. So, we can take this outside of the square root we have,
So, \[\sqrt {98} = 7 \times \sqrt 2 \]
Since $2$ is not a perfect square, we can multiply this and keep it inside the square root,
$ \Rightarrow \sqrt {98} = 7\sqrt 2 $
This is the simplified form of $\sqrt {98} $.
Now, we also have $\sqrt 2 $. $2$ is a prime number as it has only two factors, one and the number itself. So, $\sqrt 2 $ cannot be simplified further.
Hence, we have the expression as $\sqrt {98} \div \sqrt 2 = \dfrac{{\sqrt {98} }}{{\sqrt 2 }} = \dfrac{{7\sqrt 2 }}{{\sqrt 2 }}$
Now, we cancel the common factors in numerator and denominator. So, we get,
$ \Rightarrow 7$
Hence, the value of $\sqrt {98} \div \sqrt 2 $ is equal to $7$.
Therefore, option (C) is the correct answer.

Note: Here $\sqrt {} $ is the radical symbol used to represent the root of numbers. The number under the radical symbol is called radicand. The positive number, when multiplied by itself, represents the square of the number. The square root of the square of a positive number gives the original number. To find the factors, find the smallest prime number that divides the given number and divide it by that number, and then again find the smallest prime number that divides the number obtained and so on. The set of prime numbers obtained that are multiplied to each other to form the bigger number are called the factors.