In Mathematics, surds are the values in square root that cannot be further simplified into whole numbers or integers. Surds are irrational numbers. The examples of surds are √2, √3, √5, etc., as these values cannot be further simplified. If we further simply them, we get decimal values, such as:
√2 = 1.4142135…
√3 = 1.7320508…
√5 = 2.2360679…
| Also, read: |
Surds Definition
Surds are the square roots (√) of numbers that cannot be simplified into a whole or rational number. It cannot be accurately represented in a fraction. In other words, a surd is a root of the whole number that has an irrational value. Consider an example, √2 ≈ 1.414213. It is more accurate if we leave it as a surd √2.
Types of Surds
The different types of surds are as follows:
- Simple Surds – A surd that has only one term is called a simple surd. Example: √2, √5, …
- Pure Surds – Surds which are completely irrational. Example: √3
- Similar Surds – The surds having the same common surds factor
- Mixed Surds – Surds that are not completely irrational and can be expressed as a product of a rational number and an irrational number
- Compound Surds – An expression which is the addition or subtraction of two or more surds
- Binomial Surds – A surd that is made of two other surds
Six Rules for Surds
Rule 1:
Example:
To simplify √18
18 = 9 x 2 = 32 x 2, since 9 is the greatest perfect square factor of 18.
Therefore, √18 = √(32 x 2)
= √32 x √2
= 3 √2
Rule 2:
Example:
√(12 / 121) = √12 / √121
=√(22 x 3) / 11
=√22 x √3 / 11
= 2√3 / 11
Rule 3:
You can rationalize the denominator by multiplying the numerator and denominator by the denominator.
Example:
Rationalise
5/√7
Multiply numerator and denominator by √7
5/√7 = (5/√7) x (√7/√7)
= 5√7/7
Rule 4:
Example:
To simplify,
5√6 + 4√6
5√6 + 4√6 = (5 + 4) √6
by the rule
= 9√6
Rule 5:
Multiply top and bottom by a-b √n
This rule enables us to rationalise the denominator.
Example:
To Rationalise
Rule 6:
This rule enables you to rationalise the denominator.
Multiply top and bottom by a + b√n
Example:
To Rationalise
Surds and Indices
Surds are the root values that cannot be written as whole numbers. Indices are the power or exponent of a value. For example, for 32, 2 is the index and 3 is the base.
Suppose, we have a value √33. This value can be written as:
√33 = (33)½ = 33/2
Where 3/2 is the index.
Let us solve some examples here:
Example 1: 163/2 + 16-3/2 = ?
We know, by laws of exponents,
am x an = am+n
a-m = 1/am
⇒163/2 + 1/163/2
⇒(161/2)3 + 1/(161/2)3
⇒(42 x 1/2)3 + 1/(42 x 1/2)3
⇒43 + 1/43
⇒ 64 + 1/64
⇒ (64 x 64+ 1)/64
= (4096+1)/64
= 4097/64
Example 2: If (1/5 )3a = 0.008 Find the value of ( 0.25)a
Solution:
(1/5 )3a = 0.008 = 8/1000 = 1/125 = (1/5 )3
Apply the Algebra Law and Solve the equation.
⇒ 3a = 3
∴ a = 1
∴ ( 0.25 )a = ( 0.25 )1 = 0.25
How to Solve Surds?
You need to follow some rules to solve expressions that involve surds. One method is to rationalize the denominators, which helps to eject the surd in the denominator. Sometimes it may be mandatory to find the greatest perfect square factor to solve surds.
Problems and Solutions
Example 1: Write down the conjugate of 5√3 + √2
Solution:
The conjugate of 5√3 + √2 is 5√3 – √2.
Example 2: Rationalise the denominator: 1/[(8√11 )- (7√5)]
Solution:
Given: 1/[(8√11 )- (7√5)]
It is known that the conjugate of (8√11 )- (7√5) is (8√11 )+(7√5)
To rationalize the denominator of the given fraction, multiply the conjugate of denominator on both numerator and denominator.
=[1/[(8√11 )- (7√5)]]× [[(8√11 )+ (7√5)]/[(8√11 )+(7√5)]]
=[(8√11 )+ (7√5)]/[(8√11 )2-(7√5)2]
=[(8√11 )+ (7√5)]/[704- 245]
= [(8√11 )+ (7√5)]/459
Example 3: Multiply √7 x √2
Solution: √7 x √2 = √(7 x 2) = √14
Example 4: Divide √10 by √5.
Solution: √10/√5 = √(10/5) = √2
Example 5: Solve √x + 2√x.
Solution: √x + 2√x
= 3√x
Practice Questions
- 2√2 x 3√5 [Answer: 6√10]
- Write √28 in simplest form.
- Solve (√10 + √3) (√10 – √3)
- Solve by rationalising (5/(3 + √5).
- Write in simplest form: 3/√5
Now to start practising problems and examples of surd based on rules mentioned above, please visit BYJU’S – The Learning App.
is this surd or not : sqrt ( 3+ (sqrt 2) ) . Give proper reason.
sqrt ( 3+ (sqrt 2) ) is a surd because it cannot be simplified into a whole or rational number.
Hi mj
It is a surd
cuz it is an irrational number
This is a surd as it is an irrational number. Rational numbers have either the terminating decimal places or non terminating but repeating decimal places. sqrt ( 3+ (sqrt 2) ) the decimal places of the answer never terminates and never repeats, thus it is an irrational number and a surd
√800+√200-2√32
√800+√200-2√32
800 = 2x2x2x2x2x5x5
200 = 2x2x2x5x5
32 = 2x2x2x2x2
Therefore, taking out the terms, under the root, that are in pair of two, we get:
= 20√2 + 10√2 – (2 x 4√2)
= 20√2 + 10√2 – 8√2
= 22√2
Please help me solve this question
5√6+3√24-6√54+√150
5√6+3√24-6√54+√150
= 5√6+3.4√6-6.3√6+5√6
= 5√6+12√6-18√6+5√6
= (5+12-6+5)√6
= 16√6
5√6+3√24-6√54+√150
=5√6+3√4*6-6√9*6+√25*6
=5√6+3*2√6-6*3√6+5√6
=5√6+6√6-18√6+5√6
=-2√6
5root6+6root6-18root6+5root6
16root6-18root6
=-2root6
Thanks so much we really appreciate
Rationalize the denominator:
2/[9√13)-(8√5)]
2/[9√13)-(8√5)] = [2(9√13+8√5)]/733
Thanks to you all I really like this lesson
Can you help me with this
2✓2 + 3✓3/4✓3 – 2✓2
(2✓2 + 3✓3)/(4✓3 – 2✓2)
Multiply and divide by (4✓3 + 2✓2) to rationalise the denominator.
[(2✓2 + 3✓3) (4✓3 + 2✓2)]/[(4✓3)^2 – (2✓2)^2]
[8✓6 + 8 + 36 + 6✓6]/[16*3 – 4*2]
[14✓6 + 44]/40
[7✓6+22]/20