Question

How do you simplify square root of $187$ divided by square root of $17$?

Answer 1

Hint: Given the numbers in the form of a fraction. First, we will find the factors of the terms at the denominator and the numerator. Then, cancel out the common factors at the numerator and denominator. Then, write the result in a simplified form. Also, we will try to factor the radicand further. If the factors of the term are not possible, then the result in simplified form.

Formula used: The division property of the radicals is given as:
$\dfrac{{\sqrt x }}{{\sqrt y }} = \sqrt {\dfrac{x}{y}} $

Complete step-by-step solution:
First, we will write the given expression.
$ \Rightarrow \dfrac{{\sqrt {187} }}{{\sqrt {17} }}$
Now, we will apply the division property of the radicals to the expression.
$ \Rightarrow \sqrt {\dfrac{{187}}{{17}}} $
Now, find the factors of the numerator and denominator of the expression.
$ \Rightarrow 187 = 1 \times 11 \times 17$
$ \Rightarrow 17 = 1 \times 17$
Now, write the factors into the expression.
$ \Rightarrow \sqrt {\dfrac{{1 \times 11 \times 17}}{{1 \times 17}}} $
Now, we will cancel out the common terms in the denominator and numerator.
$ \Rightarrow \sqrt {\dfrac{{\not 1 \times 11 \times \not 17}}{{\not 1 \times \not 17}}} $
We will write the result in simplified form:
$ \Rightarrow \sqrt {11} $
Here, $\sqrt {11} $ is not a perfect square of any number which means this number is in its simplified form.

Hence the simplified form of $\dfrac{{\sqrt {187} }}{{\sqrt {17} }}$ is $\sqrt {11} $

Additional Information: The factors of a certain number are the numbers that can divide that number without leaving any remainder. Then, the number in the radical can be solved by finding the perfect squares of the number and extract the factors out of radicals. If the number is not a perfect square, then the radical expression cannot be reduced further.

Note: In such types of questions students mainly get confused in applying the formula. As they don't know which formula they have to apply. So when a certain number is divided by another number, then to simplify the expression, factorize the numbers at the numerator and denominator using the prime factorization method. Then, cancel out the common terms so that the expression is reduced to a simplified form.