Surds Simplified | Explained with Examples
Surds
In this article, we will explore
- What are Surds
- How to Simplified surds
We will also delve into how to multiply and divide surds, tackle some hard exam questions on surds, and provide practice questions with answers to solidify 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 Simplified 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.
Approximate Decimal Values:
√3 ≈ 1.732
Remember, the surd (e.g., √3) is the exact value, while the decimal is an approximation.
Simplifying Surds
- Simplifying surds involves expressing the surd in its simplest form by extracting square factors.
Steps to Simplify a Surd:
- Factorize the Number Inside the Surd into its prime factors.
- Identify Pairs of Factors: For every pair of identical factors, one factor can be taken out of the square root.
- Simplify the Expression: Multiply the factors outside the radical and leave the remaining factors inside.
Solved Example 1
Question: Simplify √50
Solution:
Step 1: Prime Factorization
50 = 2 × 5 × 5
Step 2: Identify Pairs
- Pair of 5s.
Step 3: Simplify
√50 = √(2 × 5 × 5)
- Take one 5 out:
5√2
Simplified Surd: 5√2
Multiplying Surds
Multiply surds by multiplying the numbers inside the radicals.
Rule:
√a × √b = √(a × b)
Example:
√5 × √3 = √(5 × 3) = √15
Note: You cannot multiply a surd by a regular integer under the radical.
2 × √5 ≠ √(2 × 5)
- It remains as 2√5
Dividing Surds
Divide surds by dividing the numbers inside the radicals.
Rule:
√a ÷ √b = √(a ÷ b)
Example:
√20 ÷ √2 = √(20 ÷ 2) = √10
Note: You cannot divide a surd by a regular integer under the radical.
√14 ÷ 2 remains as (√14) / 2
Solved Example 2
Question: Simplify √72
Solution:
Step 1: Prime Factorization
72 = 2 × 2 × 2 × 3 × 3
Step 2: Identify Pairs
- Pair of 2s
- Pair of 3s
Step 3: Simplify
√72 = √(2 × 2 × 2 × 3 × 3)
- Take out one 2 and one 3:
2 × 3 = 6
- Remaining inside the radical: 2
Simplified Surd: 6√2
Solved Example 3
Question: Simplify -5√60
Solution:
Step 1: Prime Factorization
60 = 2 × 2 × 3 × 5
Step 2: Identify Pairs
- Pair of 2s
Step 3: Simplify
-5√60 = -5√(2 × 2 × 3 × 5)
- Take out one 2:
-5 × 2 = -10
- Remaining inside the radical: 3 × 5 = 15
Simplified Surd: -10√15
Solved Example 4
Question: Simplify -4√200
Solution:
Step 1: Prime Factorization
200 = 2 × 2 × 2 × 5 × 5
Step 2: Identify Pairs
- Pair of 2s
- Pair of 5s
Step 3: Simplify
-4√200 = -4√(2 × 2 × 2 × 5 × 5)
- Take out one 2 and one 5:
-4 × 2 × 5 = -40
- Remaining inside the radical: 2
Simplified Surd: -40√2
Exam Questions on Surds
Solved Example 5
Question: Simplify √98
Solution:
Step 1: Prime Factorization
98 = 2 × 7 × 7
Step 2: Identify Pairs
- Pair of 7s
Step 3: Simplify
√98 = √(7 × 7 × 2)
- Take out one 7:
7√2
Simplified Surd: 7√2
Solved Example 6
Question: Simplify 3√45
Solution:
Step 1: Prime Factorization
45 = 3 × 3 × 5
Step 2: Identify Pairs
- Pair of 3s
Step 3: Simplify
3√45 = 3√(3 × 3 × 5)
- Take out one 3:
3 × 3 = 9
- Remaining inside the radical: 5
Simplified Surd: 9√5
Simplifying Surds Questions
Solved Example 7
Question: Simplify √18 + √8
Solution:
Step 1: Simplify √18
18 = 2 × 3 × 3
√18 = 3√2
Step 2: Simplify √8
8 = 2 × 2 × 2
√8 = 2√2
Step 3: Add the Simplified Terms
3√2 + 2√2 = 5√2
Simplified Surd: 5√2
Solved Example 8
Question: Simplify (√3)²
Solution:
Step 1: Write the Expression as a Product
(√3)² = √3 × √3
Step 2: Use the Property of Square Roots
√3 × √3 = 3
Simplified Surd: 3
Multiply Surds Questions
Solved Example 9
Question: Simplify √6 × √14
Solution:
Step 1: Use the Product Property of Square Roots
- Combine the square roots:
√6 × √14 = √(6 × 14)
Step 2: Multiply Inside the Square Root
6 × 14 = 84
- so we have:
√6 × √14 = √84
Step 3: Prime Factorization
84 = 2 × 2 × 3 × 7
Step 4: Identify Pairs
- Pair of 2s
Step 5: Simplify
√84 = √(2 × 2 × 3 × 7)
- Take out one 2:
2√21
Simplified Surd: 2√21
Solved Example 10
Question: Simplify (2√5)(3√10)
Solution:
Step 1: Multiply the Coefficients
- Multiply the numbers outside the square roots:
2 × 3 = 6
- This gives us:
(2√5)(3√10) = 6√(5 × 10)
Step 2: Multiply Inside the Square Root
- Multiply the numbers inside the square root:
5 × 10 = 50
So we have:
6√50
Step 3: Simplify √50 Using Prime Factorization
50 = 2 × 5 × 5
Step 4: Identify Pairs
- Pair of 5s
Step 5: Simplify
√50 = √(2 × 5 × 5)
- Take out one 5:
6 × 5√2 = 30√2
Simplified Surd: 30√2
Dividing Surds
Solved Example 11
Question: Simplify √80 ÷ √5
Solution:
Step 1: Use the Product Property of Square Roots
- Combine the square roots:
√80 ÷ √5 = √(80 ÷ 5)
Step 2: Divide Inside the Square Root
- Calculate 80 ÷ 5:
80 ÷ 5 = 16
- Now we have:
√80 ÷ √5 = √16
Step 3: Simplify √16
- Since 16 is a perfect square:
√16 = 4
Simplified Surd: 4
Solved Example 12
Question: Simplify (6√18) ÷ (3√2)
Solution:
Step 1: Divide the Coefficients
- Divide the numbers outside the square roots:
6 ÷ 3 = 2
- This gives us:
(6√18) ÷ (3√2) = 2(√18 ÷ √2)
Step 2: Use the Quotient Property of Square Roots
- Rewrite √18 ÷ √2 as a single square root:
√18 ÷ √2 = √(18 ÷ 2)
Step 3: Divide Inside the Square Root
- Calculate
18 ÷ 2 = 9
- So we have:
√(18 ÷ 2) = √9
Step 4: Simplify √9
- Since 9 is a perfect square:
√9 = 3
Step 5: Multiply the Result Now substitute back:
2 × 3 = 6
Simplified Surd: 6
Conclusion
- Understanding how to simplify surds, multiply surds, and divide surds is essential for solving various mathematical problems, especially those that appear in exams. By mastering these concepts and practicing with hard questions, you can enhance your mathematical skills and confidence.
Practice Questions and Answers on Surds
Question 1: Simplify √75
Question 2: Simplify 2√27 + 3√12
Question 3: Simplify (√8) × (√2)
Question 4: Simplify √125 ÷ √5
Question 5: Simplify 4√45 − 2√20
Question 6: Simplify (√3)³
Question 7: Simplify √32
Question 8: Simplify (5√2)(2√8)
Question 9: Simplify √18 ÷ √2
Question 10: Simplify 3√50 + 2√8
Solutions
Question 1: Simplify √75
Step 1: Prime Factorize 75
75 = 25 × 3 = 5 × 5 × 3
Step 2: Identify Pairs
- We have a pair of 5’s.
Step 3: Simplify
- Take one 5 out of the square root:
√75 = 5√3
Answer: 5√3
Question 2: Simplify 2√27 + 3√12
Step 1: Simplify Each Square Root
- For √27: 27 = 9 × 3, so √27 = √(9 × 3) = 3√3
- For √12: 12 = 4 × 3, so √12 = √(4 × 3) = 2√3
Step 2: Multiply by the Coefficients
- 2√27 = 2 × 3√3 = 6√3.
- 3√12 = 3 × 2√3 = 6√3.
Step 3: Combine Like Terms
6√3 + 6√3 = 12√3
Answer: 12√3
Question 3: Simplify (√8) × (√2)
Step 1: Use the Product Property of Square Roots
√8 × √2 = √(8 × 2) = √16
Step 2: Simplify the Square Root
√16 = 4
Answer: 4
Question 4: Simplify √125 ÷ √5
Step 1: Use the Quotient Property of Square Roots
√125 ÷ √5 = √(125 ÷ 5) = √25
Step 2: Simplify the Square Root
√25 = 5
Answer: 5
Question 5: Simplify 4√45 − 2√20
Step 1: Simplify Each Square Root
- For √45: 45 = 9 × 5, so √45 = √(9 × 5) = 3√5
- For √20: 20 = 4 × 5, so √20 = √(4 × 5) = 2√5
Step 2: Multiply by the Coefficients
- 4√45 = 4 × 3√5 = 12√5
- 2√20 = 2 × 2√5 = 4√5
Step 3: Subtract Like Terms
12√5 − 4√5 = 8√5
Answer: 8√5
Question 6: Simplify (√3)³
Step 1: Rewrite as a Product of Square Roots
(√3)³ = √3 × √3 × √3
Step 2: Simplify Pairs of Square Roots
- Combine two of the √3 terms:
√3 × √3 = 3
- Now we have 3 × √3
Step 3: Multiply
3 × √3 = 3√3
Answer: 3√3
Question 7: Simplify √32
Step 1: Prime Factorize 32
32 = 2 × 2 × 2 × 2 × 2
Step 2: Identify Pairs
- We have two pairs of 2’s.
Step 3: Simplify
Take out two 2’s from the square root:
√32 = 4√2
Answer: 4√2
Question 8: Simplify (5√2)(2√8)
Step 1: Multiply the Coefficients
5 × 2 = 10
- so we have:
(5√2)(2√8) = 10√(2 × 8)
Step 2: Multiply Inside the Square Root
2 × 8 = 16
- so we have:
10√16
Step 3: Simplify √16
- Since √16 = 4,
- we get:
10 × 4 = 40
Answer: 40
Question 9: Simplify √18 ÷ √2
Step 1: Use the Quotient Property of Square Roots
√18 ÷ √2 = √(18 ÷ 2) = √9
Step 2: Simplify √9
- Since √9 = 3,
- we have:
√18 ÷ √2 = 3
Answer: 3
Question 10: Simplify 3√50 + 2√8
Step 1: Simplify Each Square Root
- For √50: 50 = 25 × 2, so √50 = 5√2
- For √8: 8 = 4 × 2, so √8 = 2√2
Step 2: Multiply by the Coefficients
- 3√50 = 3 × 5√2 = 15√2
- 2√8 = 2 × 2√2 = 4√2
Step 3: Combine Like Terms
15√2 + 4√2 = 19√2
Answer: 19√2