Question

How do you combine \[\sqrt{3}+2\]?

Answer 1

Hint: This question is from the topic of algebra. In this question, we will first understand about the terms base and index. In solution of this question, we will understand if we can combine the terms \[\sqrt{3}+2\] or not. If we can, then we will try to combine them. And, if we cannot combine them, then we understand from some examples what type of term can be combined.

Complete step-by-step solution:
Let us solve this question.
In this question, we have asked to combine the term given in the question. The given term is \[\sqrt{3}+2\]. The question is asking us to add the terms \[\sqrt{3}\] and 2.
So, let us first understand about the terms base and index.
We can write the term \[\sqrt{3}\] as \[{{3}^{\dfrac{1}{2}}}\] and 2 as \[{{2}^{1}}\].
Here, in the term \[{{3}^{\dfrac{1}{2}}}\], we can say that the number 3 is base and the term \[\dfrac{1}{2}\] which is in power is called index.
Similarly, in the term \[{{2}^{1}}\], we can say that the term 2 is called base and the term 1 is called the index.
Now, let us understand if we can combine the term \[\sqrt{3}+2\] (or we can say \[{{3}^{\dfrac{1}{2}}}+{{2}^{1}}\]) or not.
We cannot add the terms whose bases and indices (plural form of index) are different.
In the terms \[{{3}^{\dfrac{1}{2}}}\] and \[{{2}^{1}}\], we can see that the bases are not equal. So, we cannot add them.


Note: We should have a better knowledge in the topic of algebra to solve this type of question easily. We should know that we can combine or add or subtract the numbers only if the bases and indices are equal.
Let us understand this from the following examples:
\[a{{x}^{m}}+b{{x}^{m}}=\left( a+b \right){{x}^{m}}\]
\[{{3}^{\dfrac{1}{2}}}+{{3}^{\dfrac{1}{2}}}=2\left( {{3}^{\dfrac{1}{2}}} \right)\]
\[\sqrt{4}+6\sqrt{4}=7\sqrt{4}\]
\[5\times {{9}^{2}}+3\times {{9}^{2}}=8\times {{9}^{2}}\]
\[5\times {{3}^{3}}+3\times {{3}^{2}}=5\times {{3}^{3}}+{{3}^{3}}=6\times {{3}^{3}}\]