How to Multiply Square Roots: 4 Ways

The multiplication of square roots is mostly done by the formula √a ×√b = √(a × b) where a and b are real numbers. In this post, we will learn the 4 ways how to multiply square roots with examples.

Method 1: Multiplying Square Roots Without Coefficients

First, multiply the numbers inside the square root (called the radicand). This way we will get only one square root, and then will simplify it to get the final answer. Let us understand this with an example.

Multiply √2 and √6.

√2 × √6

= √(2 × 6)

= √12

Now, express 12 as a product of two numbers at least one of them should be a perfect square number. Note that 12= 4 × 3 (here 4 is a perfect square number as 4=2²). Thus,

√12

= √(4 × 3)

= √4 ×√3

= 2 ×√3

= 2√3

So 2√3 is the product of root 2 and root 6.

Method 2: Multiplying Square Roots With Coefficients

If we have to multiply two square roots with coefficients, then first we need to multiply the coefficients and then we will multiply the square roots left by the above Method 1. The general rule is given by m√a ×n√b = mn√(a × b), where m and n are the coefficients of the square roots.

Let us multiply 3√2 and 2√6.

We have 3√2 × 2√6

= (2×3) (√2 × √6)

= 6 (√2 × √6)

(√2 × √6) can be multiplied as before and we get (√2 × √6) = 2√3.

Now, from above we have

3√2 × 2√6

= 6 (√2 × √6) = 6 × 2√3 = 12√3.

So 12√3 is the product of 3√2 and 2√6.

Method 3: Multiplying Square Roots having Perfect Radicands

For example, both 16 and 25 are perfect squares as they are squares of 4 and 5 respectively. Therefore, √16 and √25 are square roots having perfect radicands. Their product will be computed as follows:

√16 × √25

= 4 × 5

= 20.

Method 4: Multiplying Square Roots by Simplifying Each

For example, let us multiply root 8 and root 12.

√8 × √12

= √(4×2) × √(4×3)

= 2√2 × 2√3

= 4√6 by the formula m√a × n√b = mn√(a × b).

Have You Read These Square Roots?

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