Question

Write the simplest rationalizing factor of the following:
A) \[\sqrt {24} \]
B) \[\sqrt {18} \]
C) \[\sqrt {72} \]
D) \[\sqrt {108} \]

Answer 1

Hint:
Here, we have to find the simplest rationalizing factors for the given numbers. First we will find the factors, then among all factors we will find the simplest rationalizing factor. The factor of multiplication by which rationalization is done, is called the rationalizing factor.

Complete step by step solution:
Rational numbers are the numbers that cannot be expressed in terms of fraction.
A) \[\sqrt {24} \]
Now, by factoring the number, we get
\[ \Rightarrow \sqrt {24} = \sqrt {2 \times 2 \times 6} \]
Taking square root, we get
\[ \Rightarrow \sqrt {24} = 2\sqrt 6 \]
Thus, \[2\]and \[\sqrt 6 \]are the factors of \[\sqrt {24} \].
 Since, 2 is a rational number while \[\sqrt 6 \] is the irrational number, we can make \[\sqrt 6 \]a rational number by multiplying \[\sqrt 6 \]. \[\sqrt 6 \] is the rationalizing factor of \[\sqrt {24} \].
Therefore, the simplest rationalization factor is \[\sqrt 6 \].

B) \[\sqrt {18} \]
Now, by factoring the number, we get
\[ \Rightarrow \sqrt {18} = \sqrt {3 \times 3 \times 2} \]
Taking square root, we get
\[ \Rightarrow \sqrt {18} = 3\sqrt 2 \]
Thus, \[3\] and \[\sqrt 2 \]are the factors of \[\sqrt {18} \].
Since, 3 is a rational number while \[\sqrt 2 \] is the irrational number, we can make \[\sqrt 2 \]a rational number by multiplying \[\sqrt 2 \]. \[\sqrt 2 \] is the rationalizing factor of \[\sqrt {24} \].
Therefore, the simplest rationalization factor is \[\sqrt 2 \].

C) \[\sqrt {72} \]
Now, by factoring the number, we get
\[ \Rightarrow \sqrt {72} = \sqrt {3 \times 3 \times 2 \times 2 \times 2} \]
Taking square root, we get
\[ \Rightarrow \sqrt {72} = 6\sqrt 2 \]
Thus, \[6\] and \[\sqrt 2 \]are the factors of \[\sqrt {72} \].
Since, 6 is a rational number while \[\sqrt 2 \] is the irrational number, we can make \[\sqrt 2 \]a rational number by multiplying \[\sqrt 2 \]. \[\sqrt 2 \] is the rationalizing factor of \[\sqrt {72} \].
Therefore, the simplest rationalization factor is \[\sqrt 2 \].

D) \[\sqrt {108} \]
Now, by factoring the number, we get
\[ \Rightarrow \sqrt {108} = \sqrt {3 \times 3 \times 3 \times 2 \times 2} \]
Taking square root, we get
\[ \Rightarrow \sqrt {108} = 6\sqrt 3 \]
Thus, \[6\] and \[\sqrt 3 \]are the factors of \[\sqrt {108} \].
Since, 6 is a rational number while \[\sqrt 3 \] is the irrational number, we can make \[\sqrt 3 \]a rational number by multiplying \[\sqrt 3 \]. \[\sqrt 3 \] is the rationalizing factor of \[\sqrt {108} \].
Therefore, the simplest rationalization factor is \[\sqrt 3 \].

Note:
We know that Rationalization is a process by which radicals in the denominator of an algebraic fraction are eliminated. A radical is a symbol that represents a particular root of a number. A radical expression is defined as any expression containing a radical (√) symbol. In other words, we can say that Rationalization factor is a factor which converts an irrational number into a rational number.