1.6 Rationalising
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Learning Objectives
- Understand why we rationalise denominators
- Rationalise denominators with a single surd
- Rationalise denominators with surd expressions
- Apply rationalising techniques to solve problems
Key Concepts
Introduction to Rationalising
Rationalising is the process of eliminating surds from the denominator of a fraction.
This is done to simplify expressions and make calculations easier.
Historically, rationalising was important for manual calculations, but it remains a key technique in algebraic manipulation.
Why Rationalise Denominators?
Having surds in the denominator can make calculations and comparisons difficult.
Rationalising helps to standardize expressions, making them easier to compare and work with.
It's also a convention in mathematics to present answers with rational denominators when possible.
Example 1
Consider the expression 1/√2. While this is a valid expression, it's not in the standard form.
Solution:
By rationalising, we get 1/√2 × √2/√2 = √2/2, which is the standard form.
Rationalising Denominators with a Single Surd
To rationalise a denominator with a single surd (e.g., 1/√a), multiply both numerator and denominator by that surd.
This works because √a × √a = a, which eliminates the surd in the denominator.
Example 1
Rationalise the denominator: 3/√5
Solution:
Multiply both numerator and denominator by √5: 3/√5 × √5/√5 = 3√5/5
Example 2
Rationalise the denominator: 7/√3
Solution:
Multiply both numerator and denominator by √3: 7/√3 × √3/√3 = 7√3/3
Rationalising Denominators with Surd Expressions
When the denominator contains a surd expression (e.g., a + √b or a - √b), multiply both numerator and denominator by the conjugate of the denominator.
The conjugate of (a + √b) is (a - √b), and vice versa.
This works because (a + √b)(a - √b) = a² - b, which eliminates the surd.
Example 1
Rationalise the denominator: 2/(3 + √2)
Solution:
Multiply by the conjugate (3 - √2): 2/(3 + √2) × (3 - √2)/(3 - √2) = 2(3 - √2)/((3)² - (√2)²) = 2(3 - √2)/(9 - 2) = 2(3 - √2)/7 = (6 - 2√2)/7
Example 2
Rationalise the denominator: 5/(√7 - 2)
Solution:
Multiply by the conjugate (√7 + 2): 5/(√7 - 2) × (√7 + 2)/(√7 + 2) = 5(√7 + 2)/((√7)² - 2²) = 5(√7 + 2)/(7 - 4) = 5(√7 + 2)/3 = (5√7 + 10)/3
Rationalising with Algebraic Expressions
The same principles apply when working with algebraic expressions containing surds.
Identify the type of surd in the denominator and apply the appropriate rationalising technique.
Example 1
Rationalise the denominator: x/(y√z)
Solution:
Multiply by √z/√z: x/(y√z) × √z/√z = x√z/(yz)
Example 2
Rationalise the denominator: a/(b + c√d)
Solution:
Multiply by the conjugate (b - c√d): a/(b + c√d) × (b - c√d)/(b - c√d) = a(b - c√d)/((b)² - (c√d)²) = a(b - c√d)/(b² - c²d)
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