What is Surd ? How do I learn Surd ? Why do I need to learn Surd || Simple Tricks in Surd ?. Need a tutorial on Surd. Get you covered.
What is Surd ?
Surd is simplifying of irrational or non-rational numbers which does not have a ratio. Example; √8 = 2√2
Let's see how we can do it.
Numbers such as 5, 2 whole number 1/3, can be express as exact fractions or ratios: 5/1, 7/3. These are called rational numbers.
How do I learn Surd ?
It's easy to learn Surd whenever you understand the rules. In every aspect of mathematics understanding is the worse problem, and that's why many students hate mathematics heartfully, that was why George polya introduce the course PROBLEM SOLVING as easy way to show how to devise a plan when complications or problems arose, and make it core for students to study it. Surd had contribute great to the satisfaction of square root problem, like in binomial, limit and continuity, and so many other side of mathematics and statistics.
Keep scrolling please to view mathematical ways of learning Surd.
Why do I need to learn Surd ?
As a student of course you should, because web developer, Mathematicians, computer scientist, itc , accountant, etc . In every aspect of your work you must encounter a square root situation. Therefore i suggest you get a on Surd.
In a situation where by numbers cannot be written as ratios are called irrational or non-rational numbers. Examples π = 3.141592...., The decimal digits continue without recurring. Other examples of irrational numbers are √3 = 1.732050...., and √28 = 5.291502
Properties Of Surd.
Let's discuss it.
By putting m = 9, and n = 4. Find which of the following pairs of expression are equal.
a) √(mn), √m × √n
b) √(m+n), √m + √n
c) √(m/n), √m/√n
d) √(m-n), √m - √n
e) 2√m, √2m
f) 3√n, √9n
√(9 * 4) = √36 = 6
√9 * √4 = 3 * 2 = 6 . a) hold
b) √(9+4) = √13 =
√9 + √4 = 3 + 2 = 5. Take note
c) √(9/4) = 3/2
√9/√4 = √9/√4 = 3/2
d) √(9 - 4) = √5
√9 - √4 = 3 - 2 = 1
e) 2√9 = first of all let's square 2 = 2^2√9 = √(9*4) = √36= 6
√(2 x 9)= √18 = 3√2 also take note. They are not the same.
f) 3√4 = 3*3 √4 = √(9*4) = √36 = 6
√(9*4) = √36 = 6
Please carefully review and understand the above expression, because it contains the first best proofs to get Surd done. If there is any problem regarding what we have given please don't hesitate to contact us.
Simplification of surds.
To simplify a surd, where possible, express the number under the square root sign as a product of two factors, one of which is perfect square.
Then simplify the Surd by taking the square root of the perfect square.
Example 1a
Simplify the following.
a) √45, b) √162, c) √(x²y).
Solution
a) √45 = √(9 x 5) = √9 x √5 = 3√5
b) √162 = √(81 x 2) = √81 x √2 = 9√2
c) √(x² y) = √x² * √y = x√y
In Example (c) we can see that the power of x cancel the square root.
Exercises on example 1a ;
1) √20 , 6) √72
2) √75. 7) √99
3) √63. 8) √200
4) √84. 9) √54
5) √32. 10) √24
Evaluation Of Surd In Normal Square Root.
Example 1b.
a) 2√5. b) 7√3
Solution
a) 2√5 is the same as √4 x √5 = √(4*5) = √20
b) 7√3 is also the same as √49 x √3 = √(49 x 3) = √147
Exercises on example 1b ;
a) 2√7, b) 5√11, c) 3√2
d) 2√3, e) 6√5, d) 3√10
f) 2√2, g) 2√3, h) 9√3
Addition And Subtraction Of Surd
Two or more surds can be added together or subtracted from one another if they are like surds. Note ;
Before addition or subtraction, the surd should first be simplified, if possible.
Example 2
a) 4√2 + 6√2
b) 3√5 - 7√5
c) 3√8 + √50
d) 2√27 + √75 - 5√12
Solution for above examples
a) 4√2 + 6√2 since the square root are like surd we have to simplify the root
(4+6)√2 = 10√2
b) 3√5 - 7√5 = (3 - 7)√5 = -4√5
c) 2√27 + √75 - 5√12
= 2 x 3√3 + 5√3 - 5 x 2√3
= 6√3 + 5√3 - 10√3
= 11√3 - 10√3
= (11 - 10)√3
= √3
Multiplication Of Surd.
If two or more surds are to be multiplied together. we suggest you check if they are well simplified, if they are not, simplify.
Example 3a ;
a) √27 x √50
b) √12 x 3√60 x √45
c) (2√5)²
Solution
a) √27 x √50
= √(9 x 3) x √(25 x 2)
= 3√3 x 5√2
= 15√6
b)√(4 x 3) x 3(2√15) x √(9 x 5)
= 2√3 x 6√15 x 3√5
= 36√(3 x 15 x 5)
= 36√(15 x 15)
= 36√15²
= 36 x 15
= 540
c) (2√5)^2
= 2√5 x 2√5
= 2√(5 x 5)
= 2 x 5
= 10
Example 3b;
It is sometimes possible to pair up surds to give a simpler result.
Simplify
a) √2 x √3 x √5 x √12 x √45 x √50
b)√3 x √6
a) √2 x √3 x √5 x √12 x √45 x √50
= √(2*3*5*12*45*50)
= √(2 x 50 x 3 x 12 x 5 x45)
= √(100 x 6 x 225)
= 10 x 6 x 15
= 900
b) √3 x √6 = √(3 x 6) = √18
= √(9 x 2)
= 3√2
Division Of Surd.
If a fraction has a surd in the denominator, it is usually best to rationalise the denominator. To rationalise the denominator means to make the denominator into rational number, Usually a whole number.
To do this, multiply the numerator and the denominator of the fraction by a surd that will make the denominator rational.
Example 4a ;
a) 6/√3. b) 7/√18. c) 5/√5
6/√3 = 6/√3 x √3/√3 = 6√3/√3√3 = 6√3/3 If we multiple.
b) 7/√18 = 7/√(9 x 2) = 7/3√2
7/3√2 x √2/√2 = 7√2/ 3√2 x√2
= 7√2 / 3 x 2
= 7√2 / 6
c) 5/√5
5/√5 x √5/√5 =
5√5/5 = √5
Example 4a;
a) √18/√2.
b) √5/√2.
c)√(16/7)
Solution using the first rule of Surd.
√18 = √(9 x 2) = 3√2/√2
3√2/√2 x √2/√2 = 6/2
= 3.
b) √5/√2
√5/√2 x √2/√2
√10/2 = √5
c)√(16/7)
= 4/√7 x √7/√7
= 4√7/7
Introducing Surd In Brackets.
When multiplying surds in brackets, use algebraic expansion.
Example (a + b)(d + c)
= ad + ac + bd + bc
a) (3√5 + 2)(√5 + 3)
b) (3√3 - 2√2)(√2 - 2√3)
c) (4√3 + √2)(4√3 - √2)
Solution
a)
= (3√5)√5 + (3√5)3 + 2(√5) + 2(3)
= 15 + 9√5 + 2√5 + 6
= 21 + 11√5
b)
= (3√3)√2 - (3√3)2√3 -(2√2)√2 + (2√2)2√3
= 3√6 - 6(3) - 2(2) + 4√6
= 3√6 - 18 - 4 + 4√6
= 3√6 - 22 + 4√6
= 7√6 - 22
c)
= (4√3)^2 - 4√6 + 4√6 + (√2)^2
= (4√3)^2 - (√2)^2
Recall difference of two square.
= 4 x 4 (3) - 2
= 16 x 3 - 2
= 48 - 2
= 46
Exercises on Surd In Brackets.
a) √2(√2 + √6)
b) 2√5(3√5 - 2√2)
c) (√6 - √5)(√8 + √15)
d) (√5 + √15)(2√3 - 1)
Conjugate Of A Binomial Surd.
An expression may contain two terms, for example ; 6 - √5, 3√2 + √3, 2√3 + 4. If one or both of the terms contains a surd we call it a Binomial Surd expression. To rationalise a Binomial Surd, we use the idea of the difference of two squares.
For example, To rationalise √a + b
We multiply it by √a - b to get (√a)² - b².
Example On Binomial Surd.
a) (2√3 - 4)(2√3 + 4)
b) (5 + 3√2)(5 - 3√2)
Solution
a) (2√3 - 4 )(2√3 + 4 )
= (2√3)² + (8√3) - (8√3) - (4)²
= (2√3 )² - 4²
= 4*3 - 16
= 12 - 16
= - 4
b) 5² - 15√2 + 15√2 - (3√2 )²
= 25 - 9*2
= 25 - 18
= 7
Binomial Surd division(quotient).
a) 2/(3√5 +4 )
b) 6/(2√2 - 1)
c) (2√3 + 2) / (2√3 - 2)
= 2/(3√5 + 4 )x (3√5 - 4) /(3√5 - 4)
= 2(3√5 - 4) / (3√5 + 4)(3√5 - 4)
= (6√5 - 8)/ [(3√5)² - (12√5) + (12√5 ) - 16]
= (6√5 - 8)/ (9 x 5) - 16
= (6√5 - 8)/ 45 - 16
= (6√5 - 8)/29
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