Question

What are the square roots of $0.0004$?

Answer 1

Hint: A number $y$ such that ${y^2}$$ = $$x$, or a number $y$ whose square (the product of multiplying the number by itself, or $y$$ \times $$y$) is $x$, is called a square root of a number $x$. For instance, since ${(4)^2}$$ = $$16$ and ${( - 4)^2}$$ = $$16$, hence $4$ and $ - 4$ are square roots of $16$.

Complete step by step solution:
For the easy understanding, firstly we will write $0.0004$ in scientific notations –
$0.0004$$ = $$4$$ \times $${10^{ = 4}}$
Now, for finding the square root of $0.0004$, we will add square root on the number, for which we have to find –
Square root of $0.0004$$ = $$\sqrt {0.0004} $
Or, we can say that,
Square root of $0.0004 = \sqrt {4 \times {{10}^{ - 4}}} $
Since, if there is any product inside the square root or square of the product, then it can be written as product of individual square roots or product of square root.
So, the above notation can be written as –
Square root of $0.0004 = \sqrt 4 \times \sqrt {{{10}^{ - 4}}} $
Now, $4$ is the product of $2$ to itself, hence its square root is $2$ and for exponential part, firstly we will write square root as the power $\dfrac{1}{2}$.
Square root of $0.0004$$ = 2 \times {({10^{ - 4}})^{\dfrac{1}{2}}}$
Now by using the property ${({x^y})^z} = {(x)^{yz}}$, we can get
${({10^{ - 4}})^{\dfrac{1}{2}}} = {(10)^{ - 4 \times \dfrac{1}{2}}}$
$ = {(10)^{ - 2}}$
Since, we got the square root of $4$ and ${10^{ - 4}}$ individually. Hence, we can write the combined square root.
The square root of $0.0004$$ = $$2 \times {10^{ - 2}}$
Or we can say that,
The square root of $0.0004 = 0.02$
We can take $ - 0.02$ also as its square root since if we square $ - 0.02$, we will get $0.0004$.

Note:
The principal square root is the nonnegative square root of any nonnegative real number $x$. The square root of any positive number is represented as $\sqrt x $ whereas the square root of any negative number is represented as $\sqrt { - x} $. Square root of any negative number is taken as a complex number.