Expanding Surds Using Double Bracket Multiplication
Expanding Surds Double Bracket
In this article, we will explore:
- How to expand surds using double bracket multiplication
- Importance of this skill for rationalizing surds and solving problems with rationalized denominators
- Why mastering this topic is crucial for surd-related exam questions
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.
Expanding Surds with Single Bracket Multiplication
Let’s begin with a simple example involving a single term outside a bracket.
Example:
Simplify: √3 × (3 + 2√3)
Step 1: Multiply √3 by each term inside the bracket individually.
- First term: √3 multiplied by 3 equals 3√3
- Second term: √3 multiplied by 2√3
- Multiply the coefficients: 1 × 2 equals 2
- Multiply the surds: √3 × √3 equals 3 (since √a × √a equals a)
- So, √3 × 2√3 equals 2 × 3 which is 6
Step 2: Combine the results.
- The expression becomes: 3√3 + 6
Expanding Surds Using Double Bracket Multiplication
Now, let’s explore double brackets, where we multiply two binomials involving surds.
Example:
Simplify: (√5 + 3) × (2√5 + 2)
We will use the FOIL method, which stands for First, Outside, Inside, Last.
Step 1: Multiply the First terms.
- √5 multiplied by 2√5
- Coefficients: 1 × 2 equals 2
- Surds: √5 × √5 equals 5
- Result: 2 × 5 equals 10
Step 2: Multiply the Outside terms.
- √5 multiplied by 2 equals 2√5
Step 3: Multiply the Inside terms.
- 3 multiplied by 2√5 equals 6√5
Step 4: Multiply the Last terms.
- 3 multiplied by 2 equals 6
Step 5: Combine like terms.
- Add the surd terms: 2√5 + 6√5 equals 8√5
- Add the constants: 10 + 6 equals 16
Final Answer: 16 + 8√5
Solved Example
Question: Simplify: (2√20 + √8) × (3√5 − 4√2)
Solution:
Step 1: First, simplify the surds in the expression.
Simplify √20:
- 20 equals 4 times 5
- √20 equals √(4 × 5) equals 2√5
Simplify √8:
- 8 equals 4 times 2
- √8 equals √(4 × 2) equals 2√2
Now, the expression becomes:
(2 × 2√5 + 2√2) × (3√5 − 4√2)
Simplify coefficients:
2 × 2√5 = 4√5
So, the expression simplifies to:
(4√5 + 2√2) × (3√5 − 4√2)
Now, apply the FOIL method.
Step 2: Multiply the First terms
4√5 multiplied by 3√5
- Coefficients: 4 × 3 = 12
- Surds: √5 × √5 = 5
- Result: 12 × 5 = 60
Step 3: Multiply the Outside terms
4√5 multiplied by (−4√2)
- Coefficients: 4 × (−4) = −16
- Surds: √5 × √2 = √10
- Result: −16√10
Step 4: Multiply the Inside terms.
2√2 multiplied by 3√5
- Coefficients: 2 × 3 = 6
- Surds: √2 × √5 = √10
- Result: 6√10
Step 5: Multiply the Last terms.
2√2 multiplied by (−4√2)
- Coefficients: 2 × (−4) = −8
- Surds: √2 × √2 = 2
- Result: −8 × 2 = −16
Step 6: Combine like terms.
Combine the constants: 60 and (−16)
- 60 − 16 = 44
Combine the surd terms: (−16√10) and 6√10
- (−16 + 6)√10 = −10√10
Final Answer: 44 − 10√10
Squaring a Binomial Involving Surds
Solved Example
Question: Simplify: (√7 − 6)²
Solution:
This is equivalent to (√7 − 6) × (√7 − 6)
Apply the FOIL method.
Step 1: Multiply the First terms.
- √7 multiplied by √7 = 7
Step 2: Multiply the Outside terms.
- √7 multiplied by (−6) = −6√7
Step 3: Multiply the Inside terms.
- (−6) multiplied by √7 = −6√7
Step 4: Multiply the Last terms.
- (−6) multiplied by (−6) = 36
Step 5: Combine like terms.
- Add the constants: 7 + 36 = 43
- ,Final Answer: 43 − 12√7
Conclusion
Conclusion Expanding surds using double bracket multiplication is a vital skill for rationalizing denominators and solving complex surd problems.
By mastering this technique, you’ll be well-prepared to tackle exam questions involving surds. Remember to:
• Simplify surds when possible.
• Apply the FOIL method systematically.
• Combine like terms carefully.
Practice Questions and Answers on Surds Expanding Double Brackets
Question 1: Simplify: (√2 + 5) × (√2 + 3)
Question 2: Simplify: (3√3 − 2) × (√3 + 4)
Question 3: Simplify: (2 + √5)²
Question 4: Simplify: (√6 − 4)(√6 + 4)
Question 5: Simplify: (5 + 2√3)(5 − 2√3)
Question 6: Simplify: (√7 + √2)(√7 − √2)
Question 7: Simplify: (3√2 + 4)(3√2 − 4)
Question 8: Simplify: (√3 + √5)²
Question 9: Simplify: (2√5 + 3√2)(2√5 − 3√2)
Question 10: Simplify: (√2 + √3)²
Solutions
Question 1: Simplify: (√2 + 5) × (√2 + 3)
Answer:
Step 1: Multiply the First terms.
√2 × √2 = 2
Step 2: Multiply the Outside terms.
√2 × 3 = 3√2
Step 3: Multiply the Inside terms.
5 × √2 = 5√2
Step 4: Multiply the Last terms.
5 × 3 = 15
Step 5: Combine like terms.
- Combine the surd terms:
3√2 + 5√2 = 8√2
- Add the constants:
2 + 15 = 17
Final Answer: 17 + 8√2
Question 2: Simplify: (3√3 − 2) × (√3 + 4)
Answer:
Step 1: Multiply the First terms.
3√3 × √3 = 9
Step 2: Multiply the Outside terms.
3√3 × 4 = 12√3
Step 3: Multiply the Inside terms.
(−2) × √3 = −2√3
Step 4: Multiply the Last terms.
(−2) × 4 = −8
Step 5: Combine like terms.
- Combine the surd terms:
12√3 − 2√3 = 10√3
- Combine the constants:
9 − 8 = 1
Final Answer: 1 + 10√3
Question 3: Simplify: (2 + √5)²
Answer:
This is equivalent to (2 + √5) × (2 + √5)
Step 1: Multiply the First terms.
2 × 2 = 4
Step 2: Multiply the Outside terms.
2 × √5 = 2√5
Step 3: Multiply the Inside terms.
√5 × 2 = 2√5
Step 4: Multiply the Last terms.
√5 × √5 = 5
Step 5: Combine like terms.
- Combine the surd terms:
2√5 + 2√5 = 4√5
- Add the constants:
4 + 5 = 9
Final Answer: 9 + 4√5
Question 4: Simplify: (√6 − 4)(√6 + 4)
Answer:
- Notice that this is in the form of (a − b)(a + b) = a² − b²
Step 1: Calculate a²
(√6)² = 6
Step 2: Calculate b²
4² = 16
Step 3: Subtract b² from a²
6 − 16 = −10
Final Answer: −10
Question 5: Simplify: (5 + 2√3)(5 − 2√3)
Answer:
Using (a + b)(a − b) = a² − b²
Step 1: Calculate a²
5² = 25
Step 2: Calculate b²
- (2√3)² equals 4 × 3 which is 12
Step 3: Subtract b² from a²
25 − 12 = 13
Final Answer: 13
Question 6: Simplify: (√7 + √2)(√7 − √2)
Answer:
Using (a + b)(a − b) = a² − b²
Step 1: Calculate a²
(√7)² = 7
Step 2: Calculate b²
(√2)² = 2
Step 3: Subtract b² from a²
7 − 2 = 5
Final Answer: 5
Question 7: Simplify: (3√2 + 4)(3√2 − 4)
Answer:
Using (a + b)(a − b) = a² − b²
Step 1: Calculate a²
- (3√2)² equals 9 × 2 which is 18
Step 2: Calculate b²
4² = 16
Step 3: Subtract b² from a²
18 − 16 = 2
Final Answer: 2
Question 8: Simplify: (√3 + √5)²
Answer:
Equivalent to (√3 + √5) × (√3 + √5)
Step 1: Multiply the First terms.
√3 × √3 = 3
Step 2: Multiply the Outside terms.
√3 × √5 = √15
Step 3: Multiply the Inside terms.
√5 × √3 = √15
Step 4: Multiply the Last terms.
√5 × √5 = 5
Step 5: Combine like terms.
- Combine the surd terms:
√15 + √15 = 2√15
- Add the constants:
3 + 5 = 8
Final Answer: 8 + 2√15
Question 9: Simplify: (2√5 + 3√2)(2√5 − 3√2)
Answer:
Using (a + b)(a − b) = a² − b²
Step 1: Calculate a²
- (2√5)² equals 4 × 5 which is 20
Step 2: Calculate b²
- (3√2)² equals 9 × 2 which is 18
Step 3: Subtract b² from a²
20 − 18 = 2
Final Answer: 2
Question 10: Simplify: (√2 + √3)²
Answer:
Equivalent to (√2 + √3) × (√2 + √3)
Step 1: Multiply the First terms.
√2 × √2 = 2
Step 2: Multiply the Outside terms.
√2 × √3 = √6
Step 3: Multiply the Inside terms.
√3 × √2 = √6
Step 4: Multiply the Last terms.
√3 × √3 = 3
Step 5: Combine like terms.
- Combine the surd terms:
√6 + √6 = 2√6
- Add the constants:
2 + 3 = 5
Final Answer: 5 + 2√6