Question

How do you simplify \[\sqrt{900{{n}^{5}}}\]?

Answer 1

Hint: In this problem, we have to find the value of the given square root. 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. For example, the principal square root of 900 is 30, which is denoted by \[\sqrt{900}=30\]. We can simplify the remaining term using the exponent rule and get the simplified form.

Complete step by step answer:
We know that the given square root to be simplified is,
\[\sqrt{900{{n}^{5}}}\]
 We know that the multiplication of roots formula is,
\[\Rightarrow \sqrt{xy}=\sqrt{x}\times \sqrt{y}\]
We can apply this formula in the above square root, we get
\[\Rightarrow \sqrt{900}\times \sqrt{{{n}^{5}}}\]
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.
 For example, the principal square root of 900 is 30, which is denoted by \[\sqrt{900}=30\], as
\[\sqrt{900}=\sqrt{{{\left( 30 \right)}^{2}}}\]
We can apply this in the given square root, we get
\[\Rightarrow 30\sqrt{{{n}^{5}}}\]
We can write the term using the exponent rule inside the root as,
\[\Rightarrow 30\sqrt{{{n}^{4}}\times {{n}^{1}}}=30\sqrt{{{\left( {{n}^{2}} \right)}^{2}}}\sqrt{n}\]
Now we can cancel the square root and the square, we get
\[\Rightarrow 30{{n}^{2}}\sqrt{n}\] .
Therefore, the simplified form of \[\sqrt{900{{n}^{5}}}\] is \[30{{n}^{2}}\sqrt{n}\].

Note:
We should remember 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 which is denoted by \[\sqrt{a}\], where \[\sqrt{{}}\] is called radical. We will also make mistakes while using the exponent rule, which we should concentrate on.