Question

What is the square root of 100000?

Answer 1

Hint: We can write 100000 as \[{{10}^{5}}\]. A square of a number ‘\[a\]’ is a number such that \[a={{x}^{2}}\], in other words, a number \[x\] whose square root of \[a\].now we have to find the square root of 100000. Square root of \[100000\] can be written as \[\sqrt{100000}\]

Complete step-by-step answer:
From the given question it is clear that we have to find the square root of \[100000\].
When we are trying to find the square root of a number, the number should be a positive number because we cannot write a negative number under the square root. If we write a negative number under the square root, we will get a complex number. But we want real numbers.
As we are given a positive number, we can write \[100000\] under the square root.
Let us consider\[a=100000\]
Now we have to find the square root of \[a=100000\]. let it be \[x\]
So, we can write
 \[\Rightarrow x=\sqrt{100000}\]
we can write 100000 as \[{{10}^{5}}\].
\[\Rightarrow x=\sqrt{{{10}^{5}}}\]
From the basic concept of mathematics, we can write
\[{{x}^{n}}={{x}^{{{n}_{1}}+{{n}_{2}}+{{n}_{3}}........+{{n}_{n}}}}\] where \[n={{n}_{1}}+{{n}_{2}}+{{n}_{3}}.............+{{n}_{n}}\]
So, we can write \[\sqrt{{{10}^{5}}}=\sqrt{{{10}^{2+2+1}}}\]
\[\Rightarrow x=\sqrt{{{10}^{2+2+1}}}\]
We can also write \[{{x}^{a+b+c}}={{x}^{a}}\times {{x}^{b}}\times {{x}^{c}}\]
\[\Rightarrow x=\sqrt{{{10}^{2}}\times {{10}^{2}}\times {{10}^{1}}}\]…………….(1)
From the basic concept of mathematics, we can also write
\[\sqrt{{{n}_{1}}\times {{n}_{2}}\times {{n}_{3}}.........\times {{n}_{n}}}=\sqrt{{{n}_{1}}}\times \sqrt{{{n}_{2}}}\times \sqrt{{{n}_{3}}}.........\times \sqrt{{{n}_{n}}}\]
So now we can write \[\sqrt{{{10}^{2}}\times {{10}^{2}}\times {{10}^{1}}}=\sqrt{{{10}^{2}}}\times \sqrt{{{10}^{2}}}\times \sqrt{{{10}^{1}}}\]

So, the equation becomes
\[\Rightarrow x=\sqrt{{{10}^{2}}}\times \sqrt{{{10}^{2}}}\times \sqrt{{{10}^{1}}}\]
On simplification we get
\[\Rightarrow x=10\times 10\times \sqrt{10}\]
Om multiplication of \[10\]and \[10\] we get \[100\]
\[\Rightarrow x=100\times \sqrt{10}\]
We know the standard value of \[\sqrt{10}=3.16227766\]
\[\Rightarrow x=10\times 10\times 3.16227766\]
On further simplification, we get
\[\Rightarrow x=316.227766\]
\[x=316.227766\] is the root of \[100000\].

Note: Students should avoid calculation mistakes. Because while doing calculations even a small calculation error can lead to major error while finding the root of \[100000\]. Students should know the applications of the square root.