The radical symbol (√) represents the root of a number. Radicals can be encountered in algebra, but also in carpentry or any other field involving geometry or the calculation of relative dimensions and distances. Two roots that have the same indices (degrees of a root) can be multiplied immediately. If the radicals do not have the same indices, it is possible to manipulate the expression to make them equal. If you want to know how to multiply radicals, with or without numerical coefficients, just follow these steps.
Steps
Method 1 of 3: Multiplying Radicals without Numerical Coefficients
Step 1. Make sure the radicals have the same index
To multiply the roots using the basic method, they must have the same index. The "index" is that very small number written just to the left of the top line of the radical symbol. If it is not expressed, the radical must be understood as a square root (index 2) and can be multiplied with other square roots. You can multiply the radicals with different indices, but it is a more advanced method and will be explained later. Here are two examples of multiplication between radicals with the same indices:
- Example 1: √ (18) x √ (2) =?
- Example 2: √ (10) x √ (5) =?
- Example 3: 3√ (3) x 3√(9) = ?
Step 2. Multiply the numbers under the root
Afterward, just multiply the numbers under the radical signs and keep them there. Here's how to do it:
- Example 1: √ (18) x √ (2) = √ (36)
- Example 2: √ (10) x √ (5) = √ (50)
- Example 3: 3√ (3) x 3√(9) = 3√(27)
Step 3. Simplify radical expressions
If you have multiplied the radicals, there is a good chance you can simplify them by finding perfect squares or cubes already in the first step or among the factors of the final product. Here's how to do it:
- Example 1: √ (36) = 6. 36 is a perfect square because it is the product of 6 x 6. The square root of 36 is simply 6.
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Example 2: √ (50) = √ (25 x 2) = √ ([5 x 5] x 2) = 5√ (2). Although 50 is not a perfect square, 25 is a factor of 50 (as its divisor) and is a perfect square. You can decompose 25 as 5 x 5 and move a 5 out of the square root sign, to simplify the expression.
Think of it like this: if you put 5 back into the radical, it's multiplied by itself and becomes 25 again
- Example 3: 3√ (27) = 3; 27 is a perfect cube, because it is the product of 3 x 3 x 3. The cube root of 27 is therefore 3.
Method 2 of 3: Multiplying Radicals with Numerical Coefficients
Step 1. Multiply the coefficients:
are the numbers outside the radical. If no coefficient is expressed, then a 1 may be implied. Multiply the coefficients together. Here's how to do it:
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Example 1: 3√ (2) x √ (10) = 3√ (?)
3 x 1 = 3
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Example 2: 4√ (3) x 3√ (6) = 12√ (?)
4 x 3 = 12
Step 2. Multiply the numbers within the radicals
After you have multiplied the coefficients, it is possible to multiply the numbers within the radicals. Here's how to do it:
- Example 1: 3√ (2) x √ (10) = 3√ (2 x 10) = 3√ (20)
- Example 2: 4√ (3) x 3√ (6) = 12√ (3 x 6) = 12√ (18)
Step 3. Simplify the product
Now you can simplify the numbers under the radicals by looking for perfect squares or submultiples that are perfect. Once you've simplified those terms, just multiply 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: Multiply Radicals with Different Indices
Step 1. Find the m.c.m
(least common multiple) of the indices. To find it, look for the smallest number that is divisible by both indices. Find the m.c.m. of the indices of the following equation: 3√ (5) x 2√(2) =?
The indices are 3 and 2. 6 is the m.c.m. of these two numbers, because it is the smallest multiple common to 3 and 2. 6/3 = 2 and 6/2 = 3. In order to multiply the radicals, both indices must be 6
Step 2. Write each expression with the new m.c.m
as an index. Here's what the expression would look like with the new indices:
6√(5?) x 6√(2?) = ?
Step 3. Find the number by which you need to multiply each original index to find the m.c.m
For expression 3√ (5), you will need to multiply the index 3 by 2 to get 6. For the expression 2√ (2), you will need to multiply the index 2 by 3 to get 6.
Step 4. Make this number the exponent of the number inside the radical
For the first expression, put the exponent 2 above the number 5. For the second, put the 3 above the 2. Here's what they look like:
- 3√(5) -> 2 -> 6√(52)
- 2√(2) -> 3 -> 6√(23)
Step 5. Multiply the internal numbers by the root
That's how:
- 6√(52) = 6√ (5 x 5) = 6√25
- 6√(23) = 6√ (2 x 2 x 2) = 6√8
Step 6. Enter these numbers under a single radical and connect them with a multiplication sign
Here is the result: 6 √ (8 x 25)
Step 7. Multiply them
6√ (8 x 25) = 6√ (200). This is the final answer. In some cases, you may be able to simplify these expressions: in our example, you would need a submultiple of 200 that could be a power to the sixth. But, in our case, it does not exist and the expression cannot be simplified further.
Advice
- Indices of the radical are another way to express fractional exponents. In other words, the square root of any number is that same number raised to the power 1/2, the cube root corresponds to the exponent 1/3 and so on.
- If a "coefficient" is separated from the radical sign by a plus or a minus, it is not a true coefficient: it is a separate term and must be handled separately from the radical. If a radical and another term are both enclosed in the same parentheses, for example, (2 + (square root) 5), you need to handle the 2 separately from (square root) 5 when doing the operations in parentheses, but doing calculations outside the brackets, you must consider (2 + (square root) 5) as a single whole.
- A "coefficient" is the number, if any, placed directly in front of the radical sign. So, for example, in the expression 2 (square root) 5, 5 is under the root and the number 2, set out, is the coefficient. When a radical and a coefficient are put together like this, it means they are multiplied by each other: 2 * (square root) 5.
