In order to add and subtract the square roots, they must have the same rooting. In other words, you can add or subtract 2√3 with 4√3 but not 2√3 with 2√5. There are many situations in which you can simplify the number under the root in order to proceed with the addition and subtraction operations.
Steps
Part 1 of 2: Understanding the Basics
Step 1. Whenever possible, simplify each value under the root
To do this, you need to factor the rooting to find at least one that is a perfect square, such as 25 (5 x 5) or 9 (3 x 3). At this point, you can extract the perfect square from the root sign and write it to the left of the radical leaving the other factors inside. For example, consider the problem: 6√50 - 2√8 + 5√12. Numbers outside the root are called coefficients and numbers under the root sign radicandi. Here's how you can go about simplifying:
- 6√50 = 6√ (25 x 2) = (6 x 5) √2 = 30√2. You factored the number "50" to find "25 x 2", you extracted the "5" of the perfect square "25" from the root and placed it to the left of the radical. The number "2" remained under the root. Now multiply "5" by "6", the coefficient that is already off the root, and you get 30.
- 2√8 = 2√ (4 x 2) = (2 x 2) √2 = 4√2. In this case you have decomposed "8" into "4 x 2", you have extracted "2" from the perfect square "4" and you have written it to the left of the radical leaving "2" inside. Now multiply "2" by "2", the number that is already outside the root, and you get 4 as the new coefficient.
- 5√12 = 5√ (4 x 3) = (5 x 2) √3 = 10√3. Break "12" into "4 x 3" and extract "2" from the perfect "4" square. Write it to the left of the root leaving "3" inside. Multiply "2" by "5", the coefficient already present outside the radical, and you get 10.
Step 2. Circle each term of the expression that has the same rooting
Once you have done all the simplifications, you will get: 30√2 - 4√2 + 10√3. Since you can only add or subtract terms with the same root, you should circle them to make them more visible. In our example these are: 30√2 and 4√2. You can think of this as subtracting and adding fractions where you can only combine those with the same denominator.
Step 3. If you are calculating a longer expression and there are many factors with common radicands, you can circle a pair, underline another, add an asterisk to the third and so on
Rewrite the terms of the expression so that it is easier to visualize the solution.
Step 4. Subtract or add the coefficients together with the same rooting
Now you can proceed with the addition / subtraction operations and leave the other parts of the equation unchanged. Do not combine the radicandi. The concept behind this operation is to write how many roots with the same rooting are present in the expression. Non-similar values must remain alone. Here's what you need to do:
- 30√2 - 4√2 + 10√3 =
- (30 - 4)√2 + 10√3 =
- 26√2 + 10√3
Part 2 of 2: Practice
Step 1. First exercise
Add the following roots: √ (45) + 4√5. Here is the procedure:
- Simplify √ (45). First factor the number 45 and you get: √ (9 x 5).
- Extract the number "3" from the perfect square "9" and write it as the coefficient of the radical: √ (45) = 3√5.
- Now add the coefficients of the two terms that have a common root and you will get the solution: 3√5 + 4√5 = 7√5
Step 2. Second exercise
Solve the expression: 6√ (40) - 3√ (10) + √5. Here's how you should proceed:
- Simplify 6√ (40). Decompose "40" into "4 x 10" and you get that 6√ (40) = 6√ (4 x 10).
- Extract "2" from the perfect square "4" and multiply it by the existing coefficient. Now you have: 6√ (4 x 10) = (6 x 2) √10.
- Multiply the coefficients together: 12√10.
- Now re-read the problem: 12√10 - 3√ (10) + √5. Since the first two terms have the same rooting, you can proceed with the subtraction, but you will have to leave the third term unchanged.
- You will get: (12-3) √10 + √5 which can be simplified to 9√10 + √5.
Step 3. Third exercise
Solve the following expression: 9√5 -2√3 - 4√5. In this case there are no radicands with perfect squares and no simplification is possible. The first and third terms have the same rooting, so they can be subtracted from each other (9 - 4). The radicandi remain the same. The second term is not similar and is rewritten as it is: 5√5 - 2√3.
Step 4. Fourth exercise
Solve the following expression: √9 + √4 - 3√2. Here is the procedure:
- Since √9 is equal to √ (3 x 3), you can simplify √9 to 3.
- Since √4 is equal to √ (2 x 2), you can simplify √4 to 2.
- Now do the simple addition: 3 + 2 = 5.
- Since 5 and 3√2 are not similar terms, there is no way to add them together. The final solution is: 5 - 3√2.
Step 5. Fifth exercise
In this case we add and subtract square roots that are part of a fraction. Just like in normal fractions, you can add and subtract only between those with a common denominator. Suppose we solve: (√2) / 4 + (√2) / 2. Here is the procedure:
- Make the terms have the same denominator. The lowest common denominator, the denominator that is divisible by both denominators "4" and "2", is "4".
- Recalculate the second term, (√2) / 2, with the denominator 4. To do this you need to multiply both the numerator and the denominator by 2/2. (√2) / 2 x 2/2 = (2√2) / 4.
- Add the numerators of the fractions together, leaving the denominator unchanged. Proceed as a normal addition of fractions: (√2) / 4 + (2√2) / 4 = 3√2) / 4.
Advice
Always simplify the radicands with a factor that is a perfect square, before starting to combine similar radicands
Warnings
- Never add or subtract non-similar radicals from each other.
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Do not combine whole numbers and radicals; eg Not it is possible to simplify 3 + (2x)1/2.
Note: "(2x) raised to 1/2" = (2x)1/2 is another way of writing "square root of (2x)".
