What are Surds?

“Surd” is a Latin word which means deaf or mute. In earlier days, Arabian mathematicians identified rational numbers and irrational numbers as audible and inaudible numbers. Surds are irrational numbers, so they were called asamm (deaf) in the Arabic language, and as surds in the Latin language. Surd is an essential topic in mathematics.

In this article, we will learn about surds, along with the definition of surds, the rules of surds, and simplifying calculations with surds, etc.

Surds Definition

Surds are the numbers that come in the form of square roots (√). They cannot be simplified into a whole number or a rational number, surds are irrational numbers. These numbers cannot be accurately represented as fractions. In other words, a surd has an irrational value and it is a root of the whole number. Surds are represented by square root symbols as shown in the image added below:

For example, √2 ≈ 1.414213, but accurately we shall leave it as a surd √2.

Types of Surds

Surds are classified into six different types namely Simple Surds, Pure Surds, Similar Surds, Mixed Surds, Compound Surds, and Binomial Surds. Let us discuss the types:

• Simple Surds: A simple surd is a surd that has only a term. For example, √2 or √5.

• Pure Surd: Completely irrational surds are called pure surds. For example, √3.

• Similar Surd: When two or more surds have the same factor, they are known as similar surds. For example, √2, 3√2, 5√2 are similar surds with a common factor ²√2.

• Mixed Surds: Mixed surds are the production of rational numbers and irrational numbers and they are not completely irrational numbers.

• Compound Surds: Compound Surds are the surds that are the addition or the subtraction of two or more surds.

• Binomial Surd: A binomial surd is a surd that is made up of two other surds.

Six Rules of Surds

The six important rules of the surds are added below:

Rule 1: √(a×b) = √a × √b

Example:

√15 = √(3×5) = √3 × √5

Rule 2: √(a/b) = √a/√b

Example:

√(20/21) = √20/√21 = 2√5/√21

Rule 3: b/√a = (b/√a) × (√a/√a) = b√a/a

Example:

= 2/√5

= (2/√5) × (√5/√5)

= 2√5/5

Rule 4: a√c ± b√c = (a ± b)c

Example:

= 5√5 ± 3√5

= (5 ± 3)√5

Rule 5: c/(a + b√n)

Example:

= 5/(4+2√3)

= 5(4-2√3) /{(4+2√3)(4-2√3)} (Multiply top and bottom by (4-2√3) )

= 5(4-2√3)/{4² – (2√3)²}

= 10(2-√3)/(16 – 12)

= 5(2-√3)/2

Rule 6: c/(a – b√n)

Example:

= 5/(4-2√3)

= 5(4+2√3) /{(4+2√3)(4-2√3)} (Multiply top and bottom by (4+2√3) )

= 5(4+2√3)/{4² – (2√3)²}

= 10(2+√3)/(16 – 12)

= 5(2+√3)/2

Properties of Surds

The following are the key properties of surds:

• Sum and difference of a rational number & a quadratic surd can not be equal to a quadratic surd. For example, a + b ≠ √c

• If p ± √q = x ± √y , then p = x and q = y

• If √(a+√b) = √c + √d , then √(a-√b) = √c – √d & vice versa.

• Surds cannot be added. For example, √a + √b ≠ √(a+b)

• Surds can not be subtracted. For example, √a – √b ≠ √(a-b)

• Surds can be multiplied. For example, √a × √b = √(a×b)

• Sureds can be divided. For example, √a/√b = √a/√b

Surds and Indices

Surds are the root values that can be written as irrational numbers. Indices are the power or exponent of a value. For example, for P^a, here a is the index and P is the base.

Suppose, we have a value √75, we can express it as follows:

= √125

= √5³

= (5³)^1/2 = 5^3/2

Here, 3/2 is the index and 5 is the base.

Laws of Indices

Laws of indices are as follows:

• P^m × Pⁿ = P^m+n

• P^m/Pⁿ = P^m-n

• (P^m)ⁿ = P^mn

• (AB)ⁿ = Aⁿ × Bⁿ

• (A/B)ⁿ = Aⁿ/Bⁿ

• P⁰ = 1

• P^-n = 1/Pⁿ

Example. Simplify it (256)^3/4

Solution:

= (256)^3/4

= (4⁴)^3/4 = (4)^4×3/4

= 4³ = 64

How to Solve Surds?

To solve surds, we have to follow the simple steps, these are as follows:

• Step 1: Under the root find a square number (a factor of the number).

• Step 2: Rewrite the surd as a product of this square number and other number, then calculate the root of the square number.

• Step 3: Repeat if the number under the root still has square factors.

Surds Examples and Solutions

Example 1: Find product of 4√5 and 2√3.

Solution:

= 4√5 × 2√3

= 4 × 2 × √5 × √3

= 8 × √(5 × 3)

= 8√15

Example 2: Rationalize the denominator 6/(8 + √2).

Solution:

= 6/(8 + √2)

= [6 × (8 – √2)]/[(8 + √2)(8 – √2)] [Multiply top and bottom by (8 – √2)]

= [6 × (8 – √2)]/(8² – √2²)

= [6 × (8 – √2)]/62

= 3 × (8 – √2)/31

Example 3: Write the expansion of (3 + 2√2)²

Solution:

= (3 + 2√2)²

= 3² + 2 * 3 * 2√2 + (2√2)²

= 9 + 12√2 + 8

= 17 + 12√2

Example 4: What is conjugate of 12√2 + 13

Solution:

Conjugate of 12√2 + 13

= 12√2 – 13

Example 5: If {(2⁴)^1/2}^k = 256, then find out value of k?

Solution:

{(2⁴)^1/2}^k = 256

(2^4×1/2)^k = 256

2^2k = 2⁸

2k = 8

k = 4

So, required value of k is 4.

Practice Questions on Surds

Q1: If (A/B)^n-1 = (B/A)^n-3, then find out the value of n.

• a) 1/2
• b) 1
• c) 2
• d) 7/2

Q2: Find the value of [{-1/2)²}^-2]^-1

• a) 16
• b) 1/16
• c) -16
• d) -1/16

Q3: Find out value of x from the equation √(10 + ∛x) = 4.

• a) 450
• b) 150
• c) 316
• d) 216

FAQs on Surds

What are 5 types of Surds?

Five types of surds are as follows:

1. Simple surds
2. Pure Surds
3. Similar Surds
4. Mixed Surds
5. Compound Surds

Is √7 a surd?

Yes, √7 is a surd and it is an irrational number.

Is √64 a surd or not?

√64 equals to 8 which is an exact value. Surds are irrational numbers. So, √64 is not a surd.