You can only subtract square roots directly if the numbers under the radical sign (the radicands) are the same. Think of it like subtracting like terms with variables.
Here's a breakdown of how to subtract square roots:
1. Simplify Each Square Root
- First, simplify each square root individually. Look for perfect square factors within the radicand. For example, simplify √8 as √(4 * 2) = 2√2.
2. Identify Like Terms (Same Radicand)
- Next, identify the terms that have the same number under the square root. For instance,
3√2and5√2are like terms because they both have√2.3√2and5√3are NOT like terms because the radicands (2 and 3) are different.
3. Subtract the Coefficients
-
Once you've identified like terms, subtract the coefficients (the numbers in front of the square root). Keep the radical part the same.
- Example:
5√3 - 2√3 = (5-2)√3 = 3√3
- Example:
4. Examples
Here are a few examples to illustrate the process:
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Example 1: Simple Subtraction
7√5 - 3√5 = (7 - 3)√5 = 4√5
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Example 2: With Simplification
√18 - √8- Simplify:
√(9 * 2) - √(4 * 2) = 3√2 - 2√2 - Subtract:
(3 - 2)√2 = 1√2 = √2
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Example 3: With Unlike Terms After Simplification
√27 - √12- Simplify:
√(9 * 3) - √(4 * 3) = 3√3 - 2√3 - Subtract:
(3 - 2)√3 = 1√3 = √3
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Example 4: Unlike Terms - Cannot Subtract
√7 - √5(Cannot be simplified further, and the radicands are different. Leave as is.)
5. Table Summary
| Expression | Simplification | Like Terms? | Subtraction | Result |
|---|---|---|---|---|
5√2 - 2√2 |
Already Simplified | Yes | (5 - 2)√2 |
3√2 |
√12 - √3 |
2√3 - √3 |
Yes | (2 - 1)√3 |
√3 |
√20 - √5 |
2√5 - √5 |
Yes | (2 - 1)√5 |
√5 |
√8 - √3 |
2√2 - √3 |
No | Cannot be subtracted | 2√2 - √3 |
In summary, subtracting square roots involves simplifying each radical, identifying like terms (square roots with the same radicand), and then subtracting the coefficients of those like terms. If the radicands are different after simplification, you cannot directly subtract the square roots.
