Question

What is the square root of 84?

Answer 1

Hint: Now to find the square root of the given number then we will first factorize the number into prime factors. Then we will try to find the square in the prime factors and hence rewrite the number. Now we will take the square root on both sides and hence take the square out of the root and simplify.

Complete step by step solution:
Now first let us understand the concept of square.
Square of a number is nothing but the number raised to power 2.
Now we know that if the power of a number is n then the number should be multiplied by itself n times. Since here the power of the number is 2 we multiply the number by itself 2 times.
Hence the square of a number is nothing but the number multiplied by itself 2 times.
Hence we have ${{x}^{2}}=x\times x$
Now let us understand the concept of square root.
Square root of a number is the inverse of the function square. Hence if we have ${{x}^{2}}=y$ then x is said to be the square root of y. Let us understand this by an example. Now we know that ${{2}^{2}}=4$ . Hence we have that the square root of y is 2. Now the square root of x is denoted by $\sqrt{x}$
Now consider the given number 84.
Let us first factorize the number into prime factors.
Hence we get, $84=2\times 2\times 3\times 7$
Hence we can say that $84={{2}^{2}}\times 21$
Now on taking square root on both sides we get,
$\Rightarrow \sqrt{84}=\sqrt{{{2}^{2}}\times 21}$
Now we know that $\sqrt{ab}=\sqrt{a}\times \sqrt{b}$ Hence we get,
$\Rightarrow \sqrt{84}=\sqrt{{{2}^{2}}}\times \sqrt{21}$
$\Rightarrow \sqrt{84}=2\sqrt{21}$
Hence the square root of 84 is $2\sqrt{21}$.

Note: Now note that we have multiplication of two negative numbers is always positive. Hence squares of negative numbers is always positive. Hence we can say that there is no way that we square a real number and get a negative number. This is why the square root of a negative number is not defined in real numbers.