Multiplication and division of square roots

What is a Square root?

The definition of square root can be stated like this: The square root of a number n is that number which when multiplied by itself gives n as the product.
For example: 1: Square root of 49 is 7. Mathematically its written using the square root symbol like this: √(49) = 7.

Example 2: Square root of 1/9 is 1/3. So, √(1/9) = 1/3
Square root of a number is represented by a square root sign over the number as follows:
√(a), that means square root of number a. Alternatively it can also be written as sqrt(a) or rad(a).
The square root symbol is also known as ‘radical’ (hence the rad(a)).

Square root of a number that is not a perfect square is represented in radical form. For example, square root of 2 is written as : sqrt(2) or √(2). Or square root of 10 is written as sqrt(10) or √(10) etc.

Radical rules

Rules for multiplying square roots: (how to multiply square roots)
1. √(a) * √(b) = √(a*b). In words, product of two radicals is equal to the radical of the product of the two numbers.
For example: √6*√7 = √(6*7) = √42

2. a√(b) * c√(d) = ac√(bd). When multiply square root terms that have both, whole number and radical components, the whole numbers outside are multiplied and the numbers under the radical are multiplied together.
For example : 2√(3) * 5√(11) = 2*5*√(3*11) = 10√(33)

Rules for dividing square roots:
1. √(a)/√(b) = √(a/b). In words, quotient of two radicals is equal to the radical of the quotient of the two numbers.
For example: √(15)/√(5) = √(15/5) = √(3)

2. a√(b)/c√(d) = (a/c)*(√(b/d). Similar to multiplication of radicals, even when dividing the whole number portions are divided separately and the radical portion is divided differently and then the product is taken.
For example: 10√(32)/5√(2) = (10/5) * (√(32/2)) = 2 * (√16) = 2*4 = 8 (because sqrt(16)= 4)