How to Add and Subtract Square Roots: 9 Steps

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How to Add and Subtract Square Roots: 9 Steps
How to Add and Subtract Square Roots: 9 Steps
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To add or subtract square roots, you will need to combine roots that have the same term as the radial. This means you can add and subtract 2√3 and 4√3, but not 2√3 and 2√5. There are many cases where it is possible to actually simplify the number within the radical so that they can be combined as terms and then add and remove square roots.

Steps

Part 1 of 2: Getting to Know the Basics

Add and Subtract Square Roots Step 1

Step 1. Simplify any term within the stem if possible

To do this, try factoring the terms to find at least one term that is a perfect square, such as 25 (5 x 5) or 9 (3 x 3). Then you can take the square root of the perfect square and write it outside the radical, leaving the remaining factor inside it. In this example, we will use the following problem: 6√50 - 2√8 + 5√12. The numbers outside the radical are the coefficients and the numbers inside are the radicands. See how to simplify each term:

  • 6√50 = 6√(25 x 2) = (6 x 5)√2 = 30√2. In this example, you factor "50" into "25 x 2" and take the "5" from the perfect root, "25", and place it outside the radical, with the "2" remaining inside it. Then you multiply "5" by "6", the number outside the radical, to get "30" as the new coefficient.
  • 2√8 = 2√(4 x 2) = (2 x 2)√2 = 4√2. In this example, you factor "8" into "4 x 2" and take the "2" from the perfect root, "4", and place it outside the radical, with the "2" inside it. Then you multiply "2" by "2", the number outside the radical, to get "4" as the new coefficient.
  • 5√12 = 5√(4 x 3) = (5 x 2)√3 = 10√3. In this example, you factor "12" into "4 x 3" and take the "2" from the perfect root, "4", and place it outside the radical, with the factor "3" inside it. Then you multiply "2" by "5", the number outside the radical, to get "10" as the new coefficient.
Add and Subtract Square Roots Step 2

Step 2. Circle any terms with equal radicands

After simplifying the terms' radicands, the equation will look like this: 30√2 - 4√2 + 10√3. Since it is only possible to add or subtract the same terms, circle the terms that have the same radical. In the example used, the terms are 30√2 and 4√2. Think of this procedure as being similar to adding or subtracting fractions, where you can only do this with terms of the same denominator.

Add and Subtract Square Roots Step 3

Step 3. If you are working with a long equation where there are multiple pairs with equal radicands, you can circle the first pair, underline the second, and put an asterisk in the third, and so on

Align the terms to make the solution easier to see.

Add and Subtract Square Roots Step 4

Step 4. Add or subtract the coefficients of terms with equal radicands

Now all you need to do is add or subtract the coefficients from the terms with equal radicands and leave any additional terms as part of the equation. Do not combine radicands. The idea is to identify how many types of radicals there are in total. Different terms may remain the same. Do the following:

  • 30√2 - 4√2 + 10√3 =
  • (30 - 4)√2 + 10√3 =
  • 26√2 + 10√3

Part 2 of 2: Practicing more

Add and Subtract Square Roots Step 5

Step 1. Example 1

In this example, add the following square root: √(45) + 4√5. Do the following:

  • Simplify √(45). First, factor to get √(9 x 5).
  • Then take the "3" from the perfect square root, "9", and make it the coefficient of the radical. So √(45) = 3√5.
  • Now, just add the coefficients of the two terms with equal radicands to get the answer. 3√5 + 4√5 = 7√5
Add and Subtract Square Roots Step 6

Step 2. Example 2

In this example, the problem is as follows: 6√(40) - 3√(10) + √5. Do the following:

  • Simplify 6√(40). First, factor the "40" to get "4 x 10", resulting in 6√(40) = 6√(4 x 10).
  • Then take the "2" from the perfect square root, "3", and multiply it by the current coefficient. Now, you have 6√(4 x 10) = (6 x 2)√10.
  • Multiply the two coefficients to get 12√10.
  • Now the problem is this: 12√10 - 3√(10) + √5. Since the first two terms have the same radicands, you can subtract the second term from the first and leave the third as-is.
  • Now the problem has changed to (12-3)√10 + √5, which can be simplified to 9√10 + √5.
Add and Subtract Square Roots Step 7

Step 3. Example 3

In this example, the problem is as follows: 9√5 -2√3 - 4√5. Here, none of the radicals have factors that are perfect squares, so simplification is not possible. The first and third terms are equal radicals, so their coefficients can already be combined (9-4). The radicand does not change. The remaining terms are not equal, so the problem can be simplified to 5√5 - 2√3.

Add and Subtract Square Roots Step 8

Step 4. Example 4

Let's say the problem is this: √9 + √4 - 3√2. Do the following:

  • Since √9 is the same as √(3 x 3), you can simplify √9 to 3.
  • Since √4 is the same as √(2 x 2), you can simplify √4 to 2.
  • Now, you can just add 3 + 2 to get 5.
  • Since 5 and 3√2 are not equal terms, there is nothing more to be done. The final answer is 5 - 3√2.
Add and Subtract Square Roots Step 9

Step 5. Example 5

Let's try adding and subtracting square roots that are part of a fraction. Now, just like a normal fraction, you can only add or subtract fractions that have the same numerator or denominator. Let's say the problem is as follows: (√2)/4 + (√2)/2. Do the following:

  • Make the terms have the same denominator. The lowest common denominator, or denominator divisible by both denominators, "4" and "2," is "4".
  • So to make the second term, (√2)/2, have the denominator 4, you'll need to multiply its numerator and denominator by 2/2. (√2)/2 x 2/2 = (2√2)/4.
  • Add the fraction numerators and keep the denominators the same. Do the same as you would when adding fractions. (√2)/4 + (2√2)/4 = 3√2)/4.

Tips

  • Always simplify any radicals that have perfect square root factors before to begin to identify and match equal radicands.

Notices

  • Never combine different radicals.
  • Never combine an integer with a radical so that: 3 + (2x)1/2 not can be simplified.

    • Note: say "half the power of (2x)" = (2x)1/2 is another way of saying "square root of (2x)".

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