Master the concepts of surds with our comprehensive guide. Learn about irrational roots, operations, rationalization, and more.
A surd is an irrational number that cannot be expressed as a simple fraction. It is typically written in the form √n, where n is a positive integer that is not a perfect square.
A surd is a number of the form:
√n where n is a positive integer and √n is irrational
Examples: √2, √3, √5, √7, etc.
Example 1
Identify which of the following are surds:
a) √4
b) √5
c) √9
d) √7
Solution:
a) √4 = 2 (not a surd, as it's rational)
b) √5 ≈ 2.236... (is a surd, as it's irrational)
c) √9 = 3 (not a surd, as it's rational)
d) √7 ≈ 2.645... (is a surd, as it's irrational)
Surds can be added, subtracted, multiplied, and divided following specific rules.
Surds can only be added or subtracted if they have the same radicand (number under the root).
a√n ± b√n = (a ± b)√n
Example 2
Simplify: 3√2 + 5√2 - 2√2
Solution:
3√2 + 5√2 - 2√2 = (3 + 5 - 2)√2
= 6√2
√a × √b = √(a × b)
a√n × b√m = ab√(n × m)
Example 3
Simplify: 2√3 × 3√2
Solution:
2√3 × 3√2 = (2 × 3)√(3 × 2)
= 6√6
√a ÷ √b = √(a/b)
a√n ÷ b√m = (a/b)√(n/m)
Example 4
Simplify: 6√8 ÷ 2√2
Solution:
6√8 ÷ 2√2 = (6/2)√(8/2)
= 3√4
= 3 × 2
= 6
Rationalization is the process of eliminating surds from the denominator of a fraction.
To rationalize a denominator containing a single surd, multiply both numerator and denominator by the surd.
a/√b = (a√b)/b
Example 5
Rationalize: 3/√5
Solution:
3/√5 = (3 × √5)/(√5 × √5)
= 3√5/5
To rationalize a denominator containing a binomial surd, multiply both numerator and denominator by the conjugate of the denominator.
Example 6
Rationalize: 1/(√3 + √2)
Solution:
1/(√3 + √2) = 1/(√3 + √2) × (√3 - √2)/(√3 - √2)
= (√3 - √2)/(3 - 2)
= √3 - √2
Conjugate pairs are expressions that differ only in the sign between terms.
For a binomial surd a + b√n, its conjugate is a - b√n
The product of a binomial surd and its conjugate is always rational.
Example 7
Find the conjugate of 2 + 3√5 and verify their product is rational.
Solution:
Conjugate = 2 - 3√5
Product = (2 + 3√5)(2 - 3√5)
= 4 - (3√5)²
= 4 - 9 × 5
= 4 - 45
= -41 (rational)
Surds can be simplified by expressing them in their simplest form.
1. √(a²b) = a√b
2. √(a/b) = √a/√b
3. √(a²) = |a|
Example 8
Simplify: √50
Solution:
√50 = √(25 × 2)
= √25 × √2
= 5√2
Example 9
Simplify: √(18/2)
Solution:
√(18/2) = √9
= 3
Surds are commonly used in various mathematical and real-world applications.
Example 10
In a right triangle, if the other two sides are 1 unit each, find the hypotenuse.
Solution:
Using Pythagorean Theorem:
h² = 1² + 1²
h² = 2
h = √2
Test your understanding of Surds with 20 fully solved, step-by-step questions designed for beginners.
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