Rationalising Surds: A Comprehensive Guide with Examples
Rationalising Surds
In this article, we will explore:
- What is rationalizing the denominator of a surd
- How to rationalize the denominator
- Importance of rationalizing denominators for simplifying expressions involving surds
- Why mastering this skill is crucial for mathematics exams
Before diving into rationalization, it’s important to have a solid understanding of adding, subtracting, multiplying, dividing, and simplifying surds. If you’re unfamiliar with these concepts, reviewing them first is recommended.
They are very important in practicing questions for Algebra as well.
Here is one more link to practice a few extra questions: Maths Genie Rationalising Surds Questions
What Is Rationalization?
- A surd is an irrational root of a rational number that cannot be simplified to remove the radical (square root) symbol.
- Rationalization involves eliminating the surd from the denominator of a fraction, making the denominator a rational number.
Why Rationalize the Denominator?
- Simplification: Expressions are considered fully simplified when the denominator is rational.
- Standard Form: Mathematical conventions prefer rational denominators for clarity and ease of further computation.
- Calculations: Rational denominators simplify the process of adding, subtracting, or comparing fractions.
Rationalizing Denominators with One Term
- When the denominator consists of a single surd, you can rationalize it by multiplying both the numerator and the denominator by that surd.
Steps:
Step 1: Identify the surd in the denominator.
Step 2:Multiply both the numerator and the denominator by this surd.
Step 3:Simplify the resulting expression.
Solved Example
Question: Rationalize the denominator of: 5/√6
Solution:
Step 1: Multiply the numerator and the denominator by √6.
- Numerator: 5 × √6 = 5√6.
- Denominator: √6 × √6 = 6.
Step 2: Write the simplified expression.
Result: 5√6/6
Now, the denominator is rational, and the expression is rationalized.
Solved Example
Question: Rationalize the denominator of: 7/√5
Solution:
Step 1: Multiply the numerator and the denominator by √5.
- Numerator: 7 × √5 = 7√5
- Denominator: √5 × √5 = 5
Step 2: Simplify the expression.
Result: 7√5/5
Solved Example
Question: Rationalize the denominator of: 3/2√3
Solution:
Step 1: Multiply the numerator and the denominator by √3.
- Numerator: 3 × √3 = 3√3
- Denominator: 2√3 × √3 = 2 × 3 = 6
Step 2: Simplify the expression.
- Result: 3√3/6
Step 3: Simplify the fraction.
Final Result: √3/2 (since 3/6 = 1/2)
Rationalizing Denominators with Two Terms (Binomials)
- When the denominator contains two terms, especially with a surd and a rational number, you need to use a different approach involving the conjugate.
What Is a Conjugate?
- The conjugate of a binomial a + b is a – b, and vice versa. Multiplying a binomial by its conjugate eliminates the surd in the denominator due to the difference of squares.
Steps:
Step 1: Identify the conjugate of the denominator.
Step 2: Multiply both the numerator and the denominator by the conjugate.
Step 3: Simplify the numerator and the denominator.
Step 4: Simplify the entire expression, if possible.
Solved Example
Question: Rationalize the denominator of: 5/(√6 + 3).
Solution:
Step 1: Identify the conjugate of the denominator.
- Conjugate: √6 – 3.
Step 2: Multiply the numerator and the denominator by the conjugate.
- Numerator: 5 × (√6 – 3) = 5√6 – 15
- Denominator: (√6 + 3)(√6 – 3) = 6 – 9 = -3
Step 3: Simplify by dividing the numerator by -3.
Final Result: -5√6/3 + 5
Solved Example
Question: Rationalize the denominator of: 2/(3 – √2)
Solution:
Step 1: Conjugate of the denominator: 3 + √2.
Step 2: Multiply the numerator and the denominator by the conjugate.
- Numerator: 2 × (3 + √2) = 6 + 2√2
- Denominator: (3 – √2)(3 + √2) = 9 – 2 = 7
Final Answer: (6 + 2√2)/7
Step-by-Step Guide to Rationalizing Binomial Denominators
Step 1: Identify the Denominator and Determine Its Conjugate:
- For a denominator of the form (a + b), the conjugate is (a – b).
- For a denominator of the form (a – b), the conjugate is (a + b).
Step 2: Multiply Numerator and Denominator by the Conjugate:
- This step ensures that the value of the fraction remains unchanged because you’re multiplying by 1 (the conjugate divided by itself).
Step 3: Apply the Difference of Squares in the Denominator:
- The product (a + b)(a – b) simplifies to a² – b²
- This step eliminates the surd from the denominator.
Step 4: Simplify the Numerator:
- Expand any brackets.
- Combine like terms if possible.
Step 5: Simplify the Entire Expression:
- Reduce fractions if possible.
- Simplify any surds in the numerator.
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