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.
- Numerator: 5 × √6 = 5√6.
- Denominator: √6 × √6 = 6.
Final Answer: 5√6 / 6
- Numerator: 7 × √5 = 7√5
- Denominator: √5 × √5 = 5
Final Answer: 7√5 / 5
- Numerator: 3 × √3 = 3√3
- Denominator: 2√3 × √3 = 2 × 3 = 6
- Result: 3√3 / 6
Final Answer: √3 / 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.
- Conjugate: √6 – 3
- Numerator: 5 × (√6 – 3) = 5√6 – 15
- Denominator: (√6 + 3)(√6 – 3) = 6 – 9 = -3
Final Answer: (-5√6 / 3) + 5
- 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
Steps:
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.
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Book a Consultation →Squaring a Binomial Involving Surds
This is equivalent to (√7 − 6) × (√7 − 6). Apply the FOIL method.
- √7 multiplied by √7 = 7
- √7 multiplied by (−6) = −6√7
- (−6) multiplied by √7 = −6√7
- (−6) multiplied by (−6) = 36
- Add the constants: 7 + 36 = 43
Final Answer: 43 − 12√7
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.
