# 3 Ways to Multiply Radicals

The radical symbol (√) represents the square root of a number. This symbol can be found in algebra, carpentry, or even an account that involves geometry or calculating relative sizes or distances. It is possible to multiply two radicals of equal indexes (degrees of a root). If they don't have the same indices, you can manipulate the equation to make this possible. Keep slow to learn how to multiply radicals with or without coefficients.

## Steps

### Method 1 of 3: Multiplying radicals without coefficients

#### Step 1. Check if the radical has the same index

This is needed to multiply them using the basic method. The "index" is the small number written to the left of the topmost line in the stem symbol. If there is no number, it is a square root (index 2), and it can be multiplied by other square roots. It is possible to multiply radicals with different indexes, but a more advanced method will be needed (see later). See two examples of multiplication using radicals with the same indices:

• Ex. 1: √(18) x √(2) = ?
• Ex. 2: √(10) x √(5) = ?
• Ex. 3: 3√(3) x 3√(9) = ?

#### Step 2. Multiply the numbers below the radical sign

Just multiply the numbers below the sign of the radical or square root and keep it there. Here's how to 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. Simplify expressions with radical

When multiplying radicals, there's a good chance you can simplify them to perfect squares or cubes, or you can simplify them by finding the perfect square as a factor in the final product. Here's how to do it:

• Ex. 1: √(36) = 6. The number 36 is a perfect square, as it is the product of the 6 x 6 multiplication. The square root of 36 is 6.
• Ex. 2: √(50) = √(25 x 2) = √([5 x 5] x 2) = 5√(2). Although the number 50 is not a perfect square, 25 is a factor of 50 (since you can divide it evenly), and it is also a perfect square. You can simplify 25 by its factors, 5 x 5, and move a 5 out of the square root sign to simplify the expression.

### Think of it this way: When you put the 5 back under the radical, it is multiplied by itself, resulting in the number 25 again

• Ex. 3:3√(27) = 3. The number 27 is a perfect cube, as it is the product of multiplying 3 x 3 x 3. Therefore, the cube root of 27 is 3.

### Method 2 of 3: Multiplying Radicals with Coefficients

#### Step 1. Multiply the coefficients

The coefficient is the number on the outside of the radical. If there is no number, the coefficient is understood to be number 1. Multiply the coefficients. Here's how to do it:

• 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. Multiply the numbers within the radicals

After multiplying the coefficients, multiply the numbers inside the radicals. Here's how to do it:

• 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 the product

Then simplify the numbers below the radicals by looking for the perfect squares by multiplying the numbers that are perfect squares. When simplifying these terms, simply multiply them by their corresponding coefficients. Here's how to do it:

• 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: Multiplying Radicals with Different Indices

#### Step 1. Find the MMC (Least Common Multiple) of the indices

To do this, find the smallest number that is equally divisible by both indices. Find the MMC of the indices of the following equation:3√(5) x 2√(2) = ?

### The indices are the numbers 3 and 2. The 6 is the MMC of these two numbers because it is the smallest number that can be equally divisible by 3 and 2. 6/3 = 2 and 6/2 = 3. To multiply the radicals, both indexes must be 6

#### Step 2. Write each expression with the new MMC as index

See how the expression will look with the new indexes:

• 6√(5) x 6√(2) = ?

#### Step 3. Find the number it would take to multiply each original index to calculate the MMC

for expression 3√(5), you need to multiply the index of 3 by 2 to get 6. For the expression 2√(2), you need to multiply the index of 2 by 3 to get 6.

#### Step 4. Make this number the exponent of the number inside the radical

For the first equation, make the number 2 the equation over the number 5. For the second equation, make the number 3 the equation over the number 2. Here's what the equations should look like:

• 2 6√(5) = 6√(5)2
• 3 6√(2) = 6√(2)3

#### Step 5. Multiply the numbers inside the radicals by their exponents

Here's how to do it:

• 6√(5)2 = 6√(5 x 5) = 6√25
• 6√(2)3 = 6√(2 x 2 x 2) = 6√8

#### Step 6. Place these numbers over a radical

Place them over a radical and connect them with a multiplication sign. See how the result will be: 6√ (8 x 25)

#### Step 7. Multiply them

6√(8 x 25) = 6√(200). That's the final answer. In some cases, it may be possible to simplify these expressions. For example, you can simplify this expression if you find a number that can be multiplied six times by itself that is a factor of 200. However, in that case the expression cannot be simplified any further.

## Tips

• If a "coefficient" is separated from the radical sign by a plus or minus sign, then it is not a coefficient; it is a separate term that must be dealt with separately from the stem. If a stem and another term are surrounded by the same parentheses - for example, (2 + √5) -, you must treat them separately when performing operations inside the parentheses, but when performing operations outside the parentheses, you must treat (2 + √5) as a whole unit.
• A radical sign is another way to identify a fractional exponent. In other words, the square root of any number is the same as that number to the 1/2 power; the cubic root of any number is the same as that number raised to the 1/3 power; and so on.
• A "coefficient" is the number, if any, placed directly in front of the radical sign. For example, in the expression (2 + √5), the number 5 is below the radical sign, and the number 2, which is outside the radical, is the coefficient. When a radical and a coefficient are put together, it is understood to be the same as multiplying the radical by the coefficient, or, continuing the previous example, 2 * √5.