How To Add Or Subtract Radicals

Adding and subtracting radicals may seem daunting at first, but with a clear understanding of the underlying principles and a step-by-step approach, it becomes a manageable task. Radicals, in their simplest form, are expressions that involve roots, such as square roots, cube roots, and so on. The key to successfully adding or subtracting them lies in identifying and combining like radicals, much like combining like terms in algebraic expressions.

Understanding Radicals

Before delving into the process of adding and subtracting radicals, it's essential to grasp the fundamental concepts.

• Radical Symbol: The radical symbol (√) indicates the root of a number. For example, √9 represents the square root of 9.
• Radicand: The radicand is the number or expression under the radical symbol. In √9, 9 is the radicand.
• Index: The index is the small number written above and to the left of the radical symbol, indicating the type of root. If no index is present, it is assumed to be 2, representing a square root. For example, ³√8 represents the cube root of 8, where 3 is the index.
• Like Radicals: Like radicals are radicals with the same index and radicand. For instance, 2√3 and 5√3 are like radicals because they both have a square root (index 2) and the same radicand (3).

Identifying Like Radicals

The ability to identify like radicals is crucial for adding and subtracting them. Like radicals must have the same index and radicand. Consider the following examples:

• Example 1: 3√5 and 7√5 are like radicals because they both have an index of 2 (square root) and the same radicand (5).
• Example 2: 4√2 and 4√3 are not like radicals because they have the same index (2) but different radicands (2 and 3).
• Example 3: 2³√7 and 5³√7 are like radicals because they both have an index of 3 (cube root) and the same radicand (7).
• Example 4: 6√11 and 6³√11 are not like radicals because they have different indices (2 and 3), even though they have the same radicand (11).

Simplifying Radicals

Often, radicals need to be simplified before they can be added or subtracted. Simplifying radicals involves breaking down the radicand into its prime factors and extracting any perfect squares (for square roots), perfect cubes (for cube roots), or higher powers that match the index of the radical.

Simplifying Square Roots

To simplify a square root, find the largest perfect square that divides the radicand evenly. For example, let's simplify √48:

1. Identify the largest perfect square that divides 48: The largest perfect square that divides 48 is 16 (since 16 x 3 = 48).
2. Rewrite the radical: √48 = √(16 x 3)
3. Apply the product rule of radicals: √(16 x 3) = √16 x √3
4. Simplify the perfect square: √16 x √3 = 4√3

So, √48 simplifies to 4√3.

Simplifying Cube Roots

To simplify a cube root, find the largest perfect cube that divides the radicand evenly. For example, let's simplify ³√54:

1. Identify the largest perfect cube that divides 54: The largest perfect cube that divides 54 is 27 (since 27 x 2 = 54).
2. Rewrite the radical: ³√54 = ³√(27 x 2)
3. Apply the product rule of radicals: ³√(27 x 2) = ³√27 x ³√2
4. Simplify the perfect cube: ³√27 x ³√2 = 3³√2

So, ³√54 simplifies to 3³√2.

Simplifying Higher Roots

The same principle applies to higher roots. For example, to simplify ⁴√32:

1. Identify the largest perfect fourth power that divides 32: The largest perfect fourth power that divides 32 is 16 (since 16 x 2 = 32).
2. Rewrite the radical: ⁴√32 = ⁴√(16 x 2)
3. Apply the product rule of radicals: ⁴√(16 x 2) = ⁴√16 x ⁴√2
4. Simplify the perfect fourth power: ⁴√16 x ⁴√2 = 2⁴√2

So, ⁴√32 simplifies to 2⁴√2.

Steps to Add or Subtract Radicals

Adding and subtracting radicals involves a few key steps:

1. Simplify each radical: Simplify each radical in the expression to its simplest form. This may involve breaking down the radicand into its prime factors and extracting any perfect squares, cubes, or higher powers.
2. Identify like radicals: Look for radicals with the same index and radicand. Only like radicals can be added or subtracted.
3. Combine like radicals: Add or subtract the coefficients (the numbers in front of the radical) of the like radicals. The radicand remains the same.
4. Write the final answer: Write the resulting expression as the sum or difference of the combined like radicals.

Adding Radicals: Examples

Let's walk through some examples of adding radicals to illustrate the process.

Example 1: Adding Simple Like Radicals

Add the following radicals: 3√5 + 7√5

1. Simplify each radical: Both radicals are already in their simplest form.
2. Identify like radicals: Both terms are like radicals because they have the same index (2) and radicand (5).
3. Combine like radicals: Add the coefficients: 3 + 7 = 10. The radicand remains the same.
4. Write the final answer: 3√5 + 7√5 = 10√5

Example 2: Adding Radicals After Simplification

Add the following radicals: √12 + √27

1. Simplify each radical:
   • √12 = √(4 x 3) = √4 x √3 = 2√3
   • √27 = √(9 x 3) = √9 x √3 = 3√3
2. Identify like radicals: Both terms are now like radicals because they have the same index (2) and radicand (3).
3. Combine like radicals: Add the coefficients: 2 + 3 = 5. The radicand remains the same.
4. Write the final answer: √12 + √27 = 2√3 + 3√3 = 5√3

Example 3: Adding Cube Roots

Add the following radicals: 2³√16 + 5³√2

1. Simplify each radical:
   • 2³√16 = 2³√(8 x 2) = 2 x ³√8 x ³√2 = 2 x 2 x ³√2 = 4³√2
   • 5³√2 is already in its simplest form.
2. Identify like radicals: Both terms are like radicals because they have the same index (3) and radicand (2).
3. Combine like radicals: Add the coefficients: 4 + 5 = 9. The radicand remains the same.
4. Write the final answer: 2³√16 + 5³√2 = 4³√2 + 5³√2 = 9³√2

Subtracting Radicals: Examples

Subtracting radicals follows a similar process as adding radicals. Let's look at some examples.

Example 1: Subtracting Simple Like Radicals

Subtract the following radicals: 9√7 - 4√7

1. Simplify each radical: Both radicals are already in their simplest form.
2. Identify like radicals: Both terms are like radicals because they have the same index (2) and radicand (7).
3. Combine like radicals: Subtract the coefficients: 9 - 4 = 5. The radicand remains the same.
4. Write the final answer: 9√7 - 4√7 = 5√7

Example 2: Subtracting Radicals After Simplification

Subtract the following radicals: √50 - √8

1. Simplify each radical:
   • √50 = √(25 x 2) = √25 x √2 = 5√2
   • √8 = √(4 x 2) = √4 x √2 = 2√2
2. Identify like radicals: Both terms are now like radicals because they have the same index (2) and radicand (2).
3. Combine like radicals: Subtract the coefficients: 5 - 2 = 3. The radicand remains the same.
4. Write the final answer: √50 - √8 = 5√2 - 2√2 = 3√2

Example 3: Subtracting Cube Roots

Subtract the following radicals: 7³√24 - 2³√3

1. Simplify each radical:
   • 7³√24 = 7³√(8 x 3) = 7 x ³√8 x ³√3 = 7 x 2 x ³√3 = 14³√3
   • 2³√3 is already in its simplest form.
2. Identify like radicals: Both terms are like radicals because they have the same index (3) and radicand (3).
3. Combine like radicals: Subtract the coefficients: 14 - 2 = 12. The radicand remains the same.
4. Write the final answer: 7³√24 - 2³√3 = 14³√3 - 2³√3 = 12³√3

Dealing with Unlike Radicals

Sometimes, after simplifying radicals, you may find that you still have unlike radicals. In such cases, you cannot combine them. The expression remains as it is. For example:

√18 + √20

1. Simplify each radical:
   • √18 = √(9 x 2) = √9 x √2 = 3√2
   • √20 = √(4 x 5) = √4 x √5 = 2√5
2. Identify like radicals: The terms 3√2 and 2√5 are unlike radicals because they have the same index (2) but different radicands (2 and 5).
3. Final answer: The expression cannot be simplified further, so the final answer is 3√2 + 2√5.

Advanced Examples

Let's tackle some more complex examples that involve multiple steps and different types of radicals.

Example 1: Combining Square Roots and Cube Roots

Simplify and combine: 2√45 - 3√20 + 4³√81 - ³√24

1. Simplify each radical:
   • 2√45 = 2√(9 x 5) = 2 x √9 x √5 = 2 x 3 x √5 = 6√5
   • 3√20 = 3√(4 x 5) = 3 x √4 x √5 = 3 x 2 x √5 = 6√5
   • 4³√81 = 4³√(27 x 3) = 4 x ³√27 x ³√3 = 4 x 3 x ³√3 = 12³√3
   • ³√24 = ³√(8 x 3) = ³√8 x ³√3 = 2³√3
2. Identify like radicals:
   • 6√5 and -6√5 are like radicals.
   • 12³√3 and -2³√3 are like radicals.
3. Combine like radicals:
   • 6√5 - 6√5 = 0√5 = 0
   • 12³√3 - 2³√3 = 10³√3
4. Write the final answer: 2√45 - 3√20 + 4³√81 - ³√24 = 0 + 10³√3 = 10³√3

Example 2: Radicals with Variables

Simplify and combine: 5√(12x³) + 2x√3x - √(75x³)

1. Simplify each radical:
   • 5√(12x³) = 5√(4 x 3 x x² x x) = 5 x √4 x √(x²) x √(3x) = 5 x 2 x x x √(3x) = 10x√(3x)
   • 2x√3x is already in its simplest form.
   • √(75x³) = √(25 x 3 x x² x x) = √25 x √(x²) x √(3x) = 5x√(3x)
2. Identify like radicals: All terms are like radicals: 10x√(3x), 2x√3x, and -5x√(3x).
3. Combine like radicals: 10x√(3x) + 2x√3x - 5x√(3x) = (10 + 2 - 5)x√(3x) = 7x√(3x)
4. Write the final answer: 5√(12x³) + 2x√3x - √(75x³) = 7x√(3x)

Common Mistakes to Avoid

When adding and subtracting radicals, it's easy to make mistakes if you're not careful. Here are some common mistakes to avoid:

• Adding or subtracting unlike radicals: Only like radicals can be combined. Ensure that the radicals have the same index and radicand before adding or subtracting them.
• Forgetting to simplify radicals: Always simplify radicals before attempting to add or subtract them. Simplifying can reveal like radicals that were not initially apparent.
• Incorrectly simplifying radicals: Make sure to correctly identify perfect squares, cubes, or higher powers when simplifying radicals. A mistake in simplification can lead to incorrect results.
• Ignoring the coefficients: Remember to add or subtract the coefficients of the like radicals, not the radicands.

Practice Problems

To solidify your understanding of adding and subtracting radicals, try these practice problems:

1. 3√8 + 5√2
2. √27 - 2√12
3. 4³√16 + ³√54
4. 2√(45x²) - x√5
5. √(28) + √(63) - √(7)

Answers:

1. 11√2
2. -√3
3. 11³√2
4. 5x√5
5. 4√7

Conclusion

Adding and subtracting radicals is a fundamental skill in algebra that requires a solid understanding of radical properties and simplification techniques. By following the steps outlined in this guide, you can confidently add and subtract radicals, whether they are simple or complex. Remember to always simplify radicals first, identify like radicals, combine them correctly, and avoid common mistakes. With practice, you'll become proficient in manipulating radicals and solving algebraic problems involving them.