2.3 Surds
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Learning Objectives
- Understand the concept of surds and irrational numbers
- Simplify surds by factorizing square roots
- Perform operations with surds (addition, subtraction, multiplication, division)
- Rationalize the denominator of surd expressions
- Solve equations involving surds
Key Concepts
Introduction to Surds
A surd is an irrational root of a number that cannot be simplified to a whole number or a fraction.
Examples of surds include √2, √3, and √5, as they are irrational numbers.
Surds are useful in mathematics as they provide exact values instead of decimal approximations.
Simplifying Surds
To simplify a surd, factorize the number under the square root into a product of a square number and another factor.
√ab = √a × √b, which allows simplification when a is a perfect square.
Example: √50 = √(25 × 2) = √25 × √2 = 5√2.
Example 1
Simplify √72.
Solution:
Step 1: Factorize 72 as 36 × 2 Step 2: Rewrite as √36 × √2 Step 3: Simplify: 6√2 So, √72 = 6√2.
Example 2
Simplify √98.
Solution:
Step 1: Factorize 98 as 49 × 2 Step 2: Rewrite as √49 × √2 Step 3: Simplify: 7√2 So, √98 = 7√2.
Operations with Surds
Addition and subtraction of surds: Surds can only be added or subtracted if they have the same radical part.
Multiplication of surds follows √a × √b = √(a × b).
Division of surds follows √a / √b = √(a / b), provided b ≠ 0.
Example 1
Simplify 3√5 + 2√5.
Solution:
Step 1: Since the surds are the same, add the coefficients: (3 + 2)√5 Step 2: Simplify: 5√5.
Example 2
Simplify √3 × √12.
Solution:
Step 1: Multiply under the root: √(3 × 12) Step 2: Simplify: √36 = 6.
Example 3
Simplify (√18) / (√2).
Solution:
Step 1: Use the property √a / √b = √(a / b) Step 2: √(18/2) = √9 = 3.
Rationalizing the Denominator
A denominator containing a surd is considered irrational. We rationalize it by multiplying both numerator and denominator by a suitable surd.
For expressions of the form 1 / √a, multiply by √a / √a.
For expressions of the form 1 / (a + √b), multiply by the conjugate (a - √b) / (a - √b).
Example 1
Rationalize 1 / √5.
Solution:
Step 1: Multiply by √5 / √5 Step 2: (1 × √5) / (√5 × √5) = √5 / 5.
Example 2
Rationalize 2 / (3 + √7).
Solution:
Step 1: Multiply by (3 - √7) / (3 - √7) Step 2: Expand: (2(3 - √7)) / ((3 + √7)(3 - √7)) Step 3: Simplify: (6 - 2√7) / (9 - 7) = (6 - 2√7) / 2 Step 4: Simplify further: 3 - √7.
Solving Equations Involving Surds
To solve equations involving surds, isolate the surd and then square both sides.
Check for extraneous solutions after squaring.
Example 1
Solve √x + 2 = 5.
Solution:
Step 1: Isolate the surd: √x = 3 Step 2: Square both sides: x = 9.
Example 2
Solve √(x - 1) = 4.
Solution:
Step 1: Square both sides: x - 1 = 16 Step 2: Solve for x: x = 17.
Test Your Knowledge
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