Question

How do you simplify \[\sqrt{21}+\sqrt{35}\]?

Answer 1

Hint: In this problem, we have to simplify the given expression. We know that every non negative real number has a non negative square root called the principal square root. We have to simplify the given principal square root. We can first separate the terms inside the square root as \[7\times 3\] and \[5\times 3\]. We can then use the multiplication of root formula and we can take the common root outside to get a simplified form:

Complete step by step answer:
We know that the given square root expression to be simplified is,
\[\sqrt{21}+\sqrt{35}\]
We can now write the above expression by separating the terms inside the root as \[7\times 3\] and \[5\times 3\], we get
\[\Rightarrow \sqrt{7\times 3}+\sqrt{7\times 5}\]
Now we can separate the terms with its individual roots by using multiplication of roots formula.
We know that the multiplication of roots formula is,
\[\sqrt{xy}=\sqrt{x}\times \sqrt{y}\]
Now we can apply this multiplication of roots formula in the above step, we get
\[\Rightarrow \sqrt{7}\times \sqrt{3}+\sqrt{7}\times \sqrt{5}\]
Now we can take the common root outside, we get
\[\Rightarrow \sqrt{7}\left( \sqrt{3}+\sqrt{5} \right)\]
Therefore, the simplified form of \[\sqrt{21}+\sqrt{35}\] is \[\sqrt{7}\left( \sqrt{3}+\sqrt{5} \right)\].

Note:
We should know that it is possible to make mistakes while writing the root formulas in which we should concentrate. We should remember that the multiplication of terms inside the root is equal to multiplication of roots with its individual terms. We can also see that all radicals are now simplified and both radicands no longer have any square root.