Question

How do you find all real square roots of $4$?

Answer 1

Hint: Here we will use the different properties of the squares and square-roots. That is $n = \sqrt n \times \sqrt n $. Here we will find the prime factors of the given number.

Complete step-by-step solution:
Here we have to find the real square root of $4$
$\sqrt 4 = \sqrt {2 \times 2} $
The above equation can be re-written as,
$\sqrt 4 = \sqrt {{2^2}} $
Square and square-root cancel each other on the right-hand side of the equation.
$ \Rightarrow \sqrt 4 = 2$
For all the real values for the square roots of $4,$
Since, the square of positive or the negative terms always gives the positive term.
$\therefore {( - 2)^2} = {(2)^2} = 4$
This is the required solution.

Additional Information: Cube is the product of same number three times such as ${n^3} = n \times n \times n$ for Example cube of $2$ is ${2^3} = 2 \times 2 \times 2$ simplified form of cubed number is ${2^3} = 2 \times 2 \times 2 = 8$. and cube-root is denoted by $\sqrt[3]{{{n^3}}} = \sqrt {n \times n \times n} = n$ For Example: $\sqrt[3]{8} = \sqrt[3]{{{2^3}}} = 2$ Do not be confused in square and square-root and similarly cubes and cube-root, know the concepts properly and apply accordingly.

Note: Prime factorization is the process of finding which prime numbers can be multiplied together to make the original number, where prime numbers are the numbers greater than $1$ and which are not the product of any two smaller natural numbers. For Example: $2,{\text{ 3, 5, 7,}}......$ $2$ is the prime number as it can have only $1$ factor.

Perfect square number is the square of an integer, simply it is the product of the same integer with itself. For example - $25{\text{ = 5 }} \times {\text{ 5, 25 = }}{{\text{5}}^2}$, generally it is denoted by n to the power two i.e. ${n^2}$. Square is the product of same number twice such as ${n^2} = n \times n$ for Example square of $2$ is ${2^2} = 2 \times 2$ simplified form of squared number is ${2^2} = 2 \times 2 = 4$ and square-root is denoted by $\sqrt {{n^2}} = \sqrt {n \times n} $ For Example: $\sqrt {{2^2}} = \sqrt 4 = 2$