# 3 ways to propagate roots

In mathematics, the symbol √ (also called radical) is that of the square root of a number. This type of symbol is found in algebra exercises, but it may be necessary to use them in everyday life, in carpentry for example or in the field of finance. When it comes to geometry, the roots are never far away! In general, we can multiply two roots provided that they have the same indices (or orders of the root). If the radicals do not have the same indices, we can try to manipulate the equation in which the roots are located so that these radicals have the same index.

## Steps

### Method 1 of 3: Multiply roots when there are no coefficients #### Step 1. First of all, check that your roots have the same index

As for classical multiplication, we must start from roots having the same index. The index is a small number written on the left side of the "root" symbol. By convention, a root without an index is a square root (of index 2). All square roots can be multiplied with each other. We can multiply roots with different indices (square roots and cubics for example), we will see that at the end of the article. Let's start with two examples of multiplication of roots having the same indices:

• Ex. 1: √ (18) x √ (2) =?
• Ex. 2: √ (10) x √ (5) =?
• Ex. 3: 3√ (3) x 3√(9) = ? #### Step 2. Multiply the radicands (numbers under the sign of the root)

Multiplying two roots (or more) of the same index amounts to multiplying the radicands (numbers under the sign of the root). This is how we do it:

• Ex. 1: √ (18) x √ (2) = √ (36)
• Ex. 2: √ (10) x √ (5) = √ (50)
• Ex. 3: 3√ (3) x 3√(9) = 3√(27) #### Step 3. Then simplify the resulting radicand

Chances are, but it is not certain, that the radicand can be simplified. In this step, we look for any perfect squares (or cubes) or we try to partially extract a perfect square from the root. See how we can proceed through these two examples:

• Ex. 1: √ (36) = 6. 36 is the perfect square of 6 (36 = 6 x 6). The square root of 36 is 6.
• Ex. 2: √ (50) = √ (25 x 2) = √ ([5 x 5] x 2) = 5√ (2). As you know, 50 is not a perfect square, but 25, which is a divisor of 50 (50 = 25 x2), is a perfect square. You can replace 25 under the root with 5 x 5. If you take 25 out of the root, a 5 goes before the root and the other disappears.

### Taken upside down, you can take your 5 and put it back under the root as long as you multiply it by itself, i.e. 25

• Ex. 3: 3√ (27) = 3. 27 the perfect cube of 3, because 27 = 3 x 3 x 3. The cube root of 27 is 3.

### Method 2 of 3: Multiply roots with coefficients #### Step 1. First multiply the coefficients

The coefficients are those numbers which affect the roots and which are to the left of the "root" sign. If there is none, the coefficient is, by convention, 1. Simply multiply the coefficients between them. Here are some examples:

• Ex. 1: 3√ (2) x √ (10) = 3√ (?)

### 3 x 1 = 3

• Ex. 2: 4√ (3) x 3√ (6) = 12√ (?)

### 4 x 3 = 12 #### Step 2. Then multiply the radicands

Once the product of the coefficients has been calculated, you can, as you saw previously, multiply the radicands. Here are some examples:

• Ex. 1: 3√ (2) x √ (10) = 3√ (2 x 10) = 3√ (20)
• Ex. 2: 4√ (3) x 3√ (6) = 12√ (3 x 6) = 12√ (18) #### Step 3. Simplify what can be simplified and do the operations

We therefore try to see if the radicand does not contain a perfect square (or cube). If this is the case, we take the root of this perfect square and we multiply it by the coefficient already present. Study the following two examples:

• 3√ (20) = 3√ (4 x 5) = 3√ ([2 x 2] x 5) = (3 x 2) √ (5) = 6√ (5)
• 12√ (18) = 12√ (9 x 2) = 12√ (3 x 3 x 2) = (12 x 3) √ (2) = 36√ (2)

### Method 3 of 3: Multiply roots with different indices #### Step 1. Determine the Least Common Multiple (PPCM) of the indices

To do this, we must find the smallest number divisible by each of the indices. Small application exercise: find the PPCM of the indices in the following expression, 3√ (5) x 2√(2) = ?

### The indices are therefore 3 and 2. 6 is the PPCM of these two numbers, because it is the smallest number divisible at the same time by 3 times and 2 (proof of this is: 6/3 = 2 and 6/2 = 3). To multiply these two roots, it will therefore be necessary to bring them back to the 6th root (expression to say "root of index 6") #### Step 2. Rewrite the expression with the roots “of index PPCM”

Here is what it looks like with our expression:

• 6√ (5) x 6√(2) = ? #### Step 3. Determine the number by which to multiply the old index to find the PPCM

For the game 3√ (5), we must multiply the index by 2 (3 x 2 = 6). For the game 2√ (2), we must multiply the index by 3 (2 x 3 = 6). #### Step 4. The indices are not changed with impunity

It is necessary to adjust the radicands. You have to raise the radicand to the power of the root multiplier. Thus, for the first part, we multiplied the index by 2, we raise the radicand to the power of 2 (square). Thus, for the second part, we multiplied the index by 3, we raise the radicand to the power of 3 (cube). Which gives us:

• 2 6√(5) = 6√(5)2
• 3 6√(2) = 6√(2)3 #### Step 5. Calculate the new radicands

This gives us:

• 6√(5)2 = 6√ (5 x 5) = 6√25
• 6√(2)3 = 6√ (2 x 2 x 2) = 6√8 #### Step 6. Multiply the two roots

As you can see, we have fallen back into the general case where the two roots have the same index. We will first reduce everything to a simple product: 6√ (8 x 25) #### Step 7. Do the multiplication:

6√ (8 x 25) = 6√ (200). This is your definitive answer. As seen previously, it may be possible that your radicand is a perfect entity. If your radicand is equal to "i" times a number ("i" being the subscript), then "i" will be your answer. Here, 200 in 6th root is not a perfect entity. We leave the answer like that.