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**Hint:**First of all, break 144 as the multiplication of small integers. We know the property that when nothing is given then the exponent of that number is 1. We also know the formula for simplification when two numbers with the same base and different exponents, \[{{x}^{a}}\times {{x}^{b}}={{x}^{a+b}}\] . Now, simplify it further and get the exponent of 2.

**Complete step-by-step solution**

According to the question, we are given a number and we have to find the exponent of 2 in its prime factorization.

Here, first of all, we need the prime factorization of 144.

We know that prime factorization is the breakdown of an integer into smaller integers. In other words, we can say that in the prime factorization method, an integer is written as the multiplication of small integers.

Also, we can say that the exponent refers to the number of times a number is multiplied by itself.

For prime factorization of 144, we need its factors.

\[\begin{align}

& 2\left| \!{\underline {\,

144 \,}} \right. \\

& 2\left| \!{\underline {\,

72 \,}} \right. \\

& 2\left| \!{\underline {\,

36 \,}} \right. \, \\

& 2\left| \!{\underline {\,

18 \,}} \right. \\

& 3\left| \!{\underline {\,

9 \,}} \right. \\

& \,\,\left| \!{\underline {\,

3 \,}} \right. \\

\end{align}\]

From the above, we can observe that

144 = \[2\times 2\times 2\times 2\times 3\times 3\] …………………………………………(1)

We know the property that when nothing is given then the exponent of that number is 1 ……………………………………(2)

We also know the formula for simplification when two numbers with the same base and different exponents, \[{{x}^{a}}\times {{x}^{b}}={{x}^{a+b}}\] …………………………………………………………………(3)

Now, from equation (1), equation (2), and equation (3), we get

144 = \[{{2}^{1}}\times {{2}^{1}}\times {{2}^{1}}\times {{2}^{1}}\times {{3}^{1}}\times {{3}^{1}}={{2}^{1+1+1+1}}\times {{3}^{1+1}}={{2}^{4}}\times {{3}^{2}}\] ……………………………………………(4)

From the above equation, we have the number which is the exponent of 2.

**Therefore, the exponent of 2 in the prime factorization of 144 is 4. Hence, the correct option is (B).**

**Note:**Here, one point should be kept into consideration. The exponent is equal to 1 when nothing is given. For instance, if we have a number a. Since nothing is mentioned here, So the exponent of a is 1.

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