Question

Evaluate the square root of \[84\] in radical form.

Answer 1

Hint: Square root of a number is the value that returns the original number multiplied by itself. Finding square root by prime factorisation is an easy method. We need factors of the number under the root and pair them in two. A pair of factor “twins” under a square root sign (radical) form a single factor outside of it.

Complete step by step answer:
We need to find the square root of \[84\] by using the prime factorization method.
Thus, the prime factorisation method of \[84\] can be written as:
First we need to divide \[84\] by the smallest prime factor i.e.\[2\]
\[84 \div 2 = 42\]
Again, number \[42\] is divided by number \[2\], we will get,
\[42 \div 2 = 21\]
Since, this number \[21\] cannot be divided by \[2\] so we will divide it by another prime number i.e. \[3\]
\[21 \div 3 = 7\]
And last, we will this number \[7\] by \[7\] itself, and we will get,
\[7 \div 7 = 1\]
Thus, the prime factorisation of \[84\] are
\[ = 2 \times 2 \times 3 \times 7\]
\[ = (2 \times 2) \times 3 \times 7\]
\[ = 4 \times 21\]
Now, we need to find the square root of the given number.
i.e. \[\sqrt {84} \]\[ = \sqrt {4 \times 21} \]
We are using the property, \[\sqrt {ab} = \sqrt a \times \sqrt b \] and we will get,
\[ = \sqrt 4 \times \sqrt {21} \]
Here, \[\sqrt {21} \] has no perfect squares as factors, so this is the most we can simplify it as below,
\[
   = 2 \times \sqrt {21} \\
   = 2\sqrt {21} \\
 \]
Hence, the square root of \[84\] in radical form is \[2\sqrt {21} \].

Note:
Alternative approach:
We need to find the factors of \[84\] into a product of perfect squares if possible and then start reducing them as much as possible.
\[\sqrt {84} = \sqrt {2 \times 2 \times 3 \times 7} \]
Factors that do not have a twin remain under the radical. Multiply them back together and leave them in there.
We are using the property, \[\sqrt {ab} = \sqrt a \times \sqrt b \] and we will get,
\[ = \sqrt {2 \times 2} \times \sqrt {3 \times 7} \]
\[ = \sqrt 4 \times \sqrt {21} \]
\[
   = 2 \times \sqrt {21} \\
   = 2\sqrt {21} \\
 \]

$\bullet $ The square root symbol is usually denoted as $\sqrt{\text{ }}$. It is called a radical symbol. The number under the radical symbol is called the radicand. The number under the radical symbol is called the radicand. i.e., \[y{\text{ }} = {\text{ }}\sqrt a \], where ‘a’ is the square of a number ‘y’. Learn square roots from 1 to 25 with some shortcut tricks here. The square root of a negative number represents a complex number.