Surds: Adding and Subtracting – A Comprehensive Guide
Surds: Adding and Subtracting
In this article, we will explore
- How to add and subtract surds
- A fundamental concept in mathematics that often appears in exams.
- Understanding surds and their manipulation is essential for solving various algebraic problems.
- We will delve into the rules governing the addition and subtraction of surds, simplify complex surds, tackle hard questions, and provide practice questions with answers to enhance your understanding.
They are very important in practicing questions for Algebra as well.
Here is one more link to practice a few extra questions: Maths Genie Surds Questions
What Are Surds?
- A surd is an irrational root of a rational number that cannot be simplified to remove the radical (square+ root) symbol.
- Surds are exact values and are left in root form because their decimal expansions are non-repeating and non-terminating.
Examples of Surds:
√3
√12
√50
These cannot be simplified to whole numbers or fractions, so they remain under the square root symbol.
Non-examples:
- √25 = 5 (since 5 squared equals 25)
- √36 = 6 (since 6 squared equals 36)
These are not surds because they simplify to rational numbers.
Adding and Subtracting Surds
The Basic Rule
- Surds can only be added or subtracted if they have the same irrational component (the same number under the square root sign).
- For example, you can add 2√7 + 4√7 because both surds contain √7
Why Can’t Different Surds Be Added Directly?
- Surds with different radicands (numbers under the square root) represent different irrational numbers. Since irrational numbers cannot be precisely calculated or compared without approximation, adding or subtracting surds with different radicands is not straightforward and generally cannot be simplified further.
Adding Surds
Rule: To add surds, they must have the same radicand. You add the coefficients (numbers in front of the surds) and keep the common surd.
Example 1
Add: 2√7 + 4√7
Solution:
- Both surds have √7
- Add the coefficients: 2 + 4 = 6
- Keep the common surd: 6√7
Example 2
Add: 5√3 + 3√3
Solution:
- Both surds have √3
- Add the coefficients: 5 + 3 = 8
- Result: 8√3
Subtracting Surds
Rule: To subtract surds, they must have the same radicand. Subtract the coefficients and keep the common surd.
Example 1
Subtract: 5√6 – 2√6
Solution:
- Both surds have √6
- Subtract the coefficients: 5 – 2 = 3
- Result: 3√6
Example 2
Subtract: 6√5 – √5
Solution:
- Remember that √5 has an implicit coefficient of 1.
- Subtract the coefficients: 6 – 1 = 5
- Result: 5√5
Simplifying Surds Before Adding or Subtracting
- Sometimes, surds need to be simplified before they can be added or subtracted. Simplifying surds involves expressing the surd in its simplest form by extracting square factors.
Steps to Simplify Surds
Step 1: Factorize the number inside the surd into its prime factors.
Step 2: Identify and extract square factors (pairs of identical factors).
Step 3: Simplify the surd by bringing out the square factors.
Solved Example 1
Question: Simplify: 5√50 – 6√2
Solution:
Step 1: Simplify √50
Prime factorization of 50:
50 = 2 × 5 × 5
√50 = √(2 × 5 × 5)
Step 2: Extract square factors
- The pair of 5s can be taken out of the square root as a single 5.
- So, √50 = 5√2
Step 3: Substitute back into the expression
5√50 = 5 × 5√2 = 25√2
Step 4: Subtract 6√2
- Both surds now have √2
- Subtract the coefficients: 25 – 6 = 19
- Result: 19√2
Solved Example 2
Question: Simplify: √20 + √45 – √12
Solution:
Step 1: Simplify each surd
Simplify √20
- Factorize: 20 = 2 × 2 × 5
- Extract the pair of 2s:
- √20 = 2√5
Simplify √45
- Factorize: 45 = 3 × 3 × 5
- Extract the pair of 3s:
- √45 = 3√5
Simplify √12
- Factorize: 12 = 2 × 2 × 3
- Extract the pair of 2s:
- √12 = 2√3
Step 2: Combine like terms
- Add 2√5 + 3√5 = 5√5
- The term 2√3 remains separate.
Final Answer: 5√5 – 2√3
Solved Example 3
Question: Simplify: √27 + √45 – √12
Solution:
Step 1: Simplify each surd
Simplify √27
- Factorize: 27 = 3 × 3 × 3
- Extract the pair of 3s:
- √27 = 3√3
Simplify √45
- Factorize: 45 = 3 × 3 × 5
- Extract the pair of 3s:
- √45 = 3√5
Simplify √12
- Factorize: 12 = 2 × 2 × 3
- Extract the pair of 2s:
- √12 = 2√3
Step 2: Combine like terms:
- Add 3√3 – 2√3 = 1√3
- The term 3√5 remains separate.
Final Answer: √27 + √45 – √12 = 1√3 + 3√5
Solved Example 4
Question: Simplify: 4√50 – 6√2
Solution:
Step 1: Simplify √50
- Prime factorization of 50:
50 = 2 × 5 × 5
√50 = √(2 × 5 × 5)
Step 2: Calculate:
4√50 = 4 × 5√2 = 20√2
- Subtract
6√2: 20√2 – 6√2 = 14√2
Answer: 14√2
Solved Example 5
Question: Simplify: √75 – √27
Solution:
Step 1: Simplify each surd
Simplify √75
- Factorize: 75 = 25 × 3
- Extract the square root of 25:
- √75 = 5√3
Simplify √27
- Factorize: 27 = 3 × 3 × 3
- Extract the pair of 3s:
- √27 = 3√3
Step 2: Subtract:
5√3 – 3√3 = 2√3
Answer: 2√3
Solved Example 6
Question: Simplify: √18 + √32 – √8
Solution:
Step 1: Simplify each surd
Simplify √18
- Factorize: 18 = 9 × 2
- Extract the square root of 9:
- √18 = 3√2
Simplify √32
- Factorize: 32 = 16 × 2
- Extract the square root of 16:
- √32 = 4√2
Simplify √8
- Factorize: 8 = 4 × 2
- Extract the square root of 4:
- √8 = 2√2
Step 2:Combine terms:
- Add 3√2 + 4√2 = 7√2
- Subtract 2√2:
7√2 – 2√2 = 5√2
Answer: 5√2
Solved Example 7
Question: Simplify: √18 + √32 – √8
Solution:
Step 1: Simplify each surd
Simplify √18
- Factorize: 18 = 9 × 2
- Extract the square root of 9:
- √18 = 3√2
Simplify √32
- Factorize: 32 = 16 × 2
- Extract the square root of 16:
- √32 = 4√2
Simplify √8
- Factorize: 8 = 4 × 2
- Extract the square root of 4:
- √8 = 2√2
Step 2:Combine terms:
- Add 3√2 + 4√2 = 7√2
- Subtract 2√2:
7√2 – 2√2 = 5√2
Answer: 5√2
Multiply Surds
Understanding how to multiply surds is essential as it often comes up in simplifying expressions.
Rule for Multiplying Surds
√a × √b = √(a × b)
Example: Multiply: √5 × √20
Solution:
- Multiply under the radical:
√(5 × 20) = √100
- Simplify:
√100 = 10
Answer: 10
Dividing Surds
Dividing surds follows a similar principle.
Rule for Dividing Surds
- √a ÷ √b = √(a ÷ b), provided b is not zero.
Example Divide: √48 ÷ √3
Solution:
- Divide under the radical:
√(48 ÷ 3) = √16
- Simplify:
√16 = 4
Answer: 4
Conclusion
Understanding how to add and subtract surds is crucial for solving various mathematical problems, especially those encountered in exams. Remember:
- Only like surds (surds with the same radicand) can be added or subtracted directly.
- Simplify surds whenever possible to identify like terms.
- When multiplying or dividing surds, use the rules for operations under radicals.
By practicing these concepts and working through various problems, you will enhance your mathematical skills and be better prepared for exam questions on surds.
Practice Questions and Answers on Surds: Adding and Subtracting
Question 1: Simplify: 3√18 + 2√8
Question 2: Simplify: √75 – √27
Question 3: Simplify: 2√20 + 3√45
Question 4: Simplify: 5√12 – 3√27
Question 5: Simplify: √98 + √18
Question 6: Simplify: 4√32 – 2√8
Question 7: Simplify: √200 – 5√8
Question 8: Simplify: 6√125 + 4√80
Question 9: Simplify: 7√12 – 2√27
Question 10: Simplify: √8 + √18 + √32
Solutions
Question 1: Simplify: 3√18 + 2√8
Solution:
Step 1: Simplify Each Surd
√18 = 3√2
- Multiply: 3√18 = 3 × 3√2 = 9√2
√8 = 2√2
- Multiply: 2√8 = 2 × 2√2 = 4√2
Step 2: Add
9√2 + 4√2 = 13√2
Answer: 13√2
Question 2: Simplify √75 – √27
Solution:
Step 1: Simplify Each Surd
√75 = 5√3
√27 = 3√3
Step 2: Subtract
5√3 – 3√3 = 2√3
Answer: 2√3
Question 3: Simplify 2√20 + 3√45
Solution:
Step 1: Simplify Each Surd
√20 = 2√5
- Multiply: 2√20 = 2 × 2√5 = 4√5
√45 = 3√5
- Multiply: 3√45 = 3 × 3√5 = 9√5
Step 2: Add
4√5 + 9√5 = 13√5
Answer: 13√5
Question 4: Simplify 5√12 – 3√27
Solution:
Step 1: Simplify Each Surd
√12 = 2√3
- Multiply: 5√12 = 5 × 2√3 = 10√3
√27 = 3√3
- Multiply: 3√27 = 3 × 3√3 = 9√3
Step 2: Subtract
10√3 – 9√3 = 1√3
Answer: 1√3
Question 5: Simplify √98 + √18
Solution:
Step 1: Simplify Each Surd
- √98 = 7√2 (since 98 = 49 × 2)
- √18 = 3√2
Step 2: Add
7√2 + 3√2 = 10√2
Answer: 10√2
Question 6: Simplify 4√32 – 2√8
Solution:
Step 1: Simplify Each Surd
√32 = 4√2
- Multiply: 4√32 = 4 × 4√2 = 16√2
√8 = 2√2
- Multiply: 2√8 = 2 × 2√2 = 4√2
Step 2: Subtract
16√2 – 4√2 = 12√2
Answer: 12√2
Question 7: Simplify √200 – 5√8
Solution:
Step 1: Simplify Each Surd
- √200 = 10√2 (since 200 = 100 × 2)
√8 = 2√2
- Multiply: 5√8 = 5 × 2√2 = 10√2
Step 2: Subtract
10√2 – 10√2 = 0
Answer: 0
Question 8: Simplify 6√125 + 4√80
Solution:
Step 1: Simplify Each Surd
√125 = 5√5
- Multiply: 6√125 = 6 × 5√5 = 30√5
√80 = 4√5
- Multiply: 4√80 = 4 × 4√5 = 16√5
Step 2: Add
30√5 + 16√5 = 46√5
Answer: 46√5
Question 9: Simplify 7√12 – 2√27
Solution:
Step 1: Simplify Each Surd
√12 = 2√3
- Multiply: 7√12 = 7 × 2√3 = 14√3
√27 = 3√3
- Multiply: 2√27 = 2 × 3√3 = 6√3
Step 2: Subtract
14√3 – 6√3 = 8√3
Answer: 8√3
Question 10: Simplify √8 + √18 + √32
Solution:
Step 1: Simplify Each Surd
- √8 = 2√2
- √18 = 3√2
- √32 = 4√2
Step 2: Add
2√2 + 3√2 + 4√2 = 9√2
Answer: 9√2