Question

How do you simplify the square root of \[{{10}^{-6}}\] ?

Answer 1

Hint: Problems of powers can be easily solved by converting all the forms (in terms of roots, reciprocals, etc.) to real number indices and then, perform the addition or subtraction or multiplication of the indices. We first convert the square root to $\dfrac{1}{2}$ as an exponent. Thereafter, we multiply the two exponents. All the required operations on indices having performed, we now reach to the simplified form.

Complete step by step answer:
Any power form of a number can be represented as
\[{{a}^{b}}\]
Where, ‘a’ is called the “base” and ‘b’ is called the “index” or “power” or “exponent”.
In the given problem, let \[x=\sqrt{{{10}^{-6}}}\] . We know that, square root is nothing but the \[\dfrac{1}{2}th\] power. Therefore, \[x\] can be simplified as,
\[x={{\left( {{10}^{-6}} \right)}^{\dfrac{1}{2}}}\]
Here, the base is \[10\] and the indices are \[-6\] and \[\dfrac{1}{2}\] . We know that, the power of a power property states that
\[{{({{a}^{b}})}^{c}}={{a}^{(b*c)}}\]
So, simplifying x accordingly, we get,
\[\begin{align}
  & x={{10}^{\left( \left( -6 \right)\times \left( \dfrac{1}{2} \right) \right)}} \\
 & \Rightarrow x={{10}^{(-3)}} \\
\end{align}\]
Now, we know that negative powers are nothing but the power of the reciprocal of the base number. In other words,
\[{{a}^{(-b)}}=\dfrac{1}{{{a}^{b}}}\]
Simplifying \[x\] according to this rule, we get,
\[\begin{align}
  & x={{10}^{(-3)}} \\
 & \Rightarrow x=\dfrac{1}{{{10}^{3}}} \\
\end{align}\]
We know that
\[{{10}^{3}}=1000\]
Thus,
\[x=\dfrac{1}{1000}\]
Writing \[\dfrac{1}{1000}\] in decimal form, we get,
\[x=0.001\]
Therefore, we can conclude that the given expression in the question can be simplified to \[x=0.001\] .

Note:
While solving power problems, we must be careful with conversion of roots and reciprocals to powers and must always remember that reciprocal is equivalent to negative power and nth root is equivalent to \[\dfrac{1}{n}th\] power. We must also be careful with the multiplication and addition of the indices, as we often get confused when to add and when to multiply. An alternate method is by writing \[{{10}^{-6}}\] as the product of two terms i.e. \[\sqrt{{{10}^{-6}}}=\sqrt{{{10}^{-3}}\times {{10}^{-3}}}\] which evaluates to \[{{10}^{-3}}\] . By doing this, \[{{10}^{-6}}\] behaves as a perfect square with its root being \[{{10}^{-3}}\] .