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Surds
SLIDE 1
Quadratic formula
If then the roots are given by
Use the formula to find the roots of , giving your answer (a) in an exact form with square root symbols, and (b) as an approximation to 2 decimal places.
SLIDE 2
exact roots approximate roots
SLIDE 3
Surds
exact roots approximate roots
An expression involving a square root symbol that cannot be removed is called a surd. All surds are therefore irrational numbers.
In the is a surd.
? Give an example of a surd that is also an irrational number.
? Give an example of a square root that is not an irrational number.
? Find the prime factorisation of 60.
? Is a rational or an irrational number?
SLIDE 4
? The square root of any prime number is an irrational number. Any surd involving a prime number is irrational. For example, , ,
? Give an example of a square root that is not an irrational number.
The square root of any square number is not an irrational number. One might say that it is a rational number masquerading as an irrational one. We unmask it by taking the square root.
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Although 4 is a factor of 60, and , is an irrational number, because it has non-square prime numbers among its factors.
SLIDE 5
Removing square numbers
Numbers appearing under a root symbol should be simplified by taking any square numbers outside the root symbol.
The square roots of prime numbers cannot be simplified.
Note
means 2 times , following the usual rule that when things are placed together unambiguously, it means the product.
SLIDE 6
Simplify by removing square numbers outside the root symbol where possible.
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SLIDE 7
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SLIDE 8
Example
Simplify in the same way
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SLIDE 9
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SLIDE 10
Example
Exercise
Collect like terms
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SLIDE 11
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SLIDE 12
Example
Note
Exercise
Expand and simplify
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SLIDE 13
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SLIDE 14
Example
Exercise
Expand and collect
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SLIDE 15
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SLIDE 16
Expand and collect
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SLIDE 17
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SLIDE 18
Example
Exercise
Expand and collect
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SLIDE 19
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SLIDE 20
Rationalising the denominator
We wish to remove a surd from the bottom (denominator) of a fraction.
Since , the way to remove from the bottom of a fraction is to multiply it by . To preserve the equation, we multiply both the top (numerator) and bottom (denominator) of the fraction by , as .
SLIDE 21
Rationalise the denominator of each of the following.
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SLIDE 22
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SLIDE 23
Conjugate
The conjugate of is because
The surd is eliminated when multiplied by the conjugate.
Example
By multiplying both numerator and denominator of by the conjugate of , rationalise the denominator.
Solution
SLIDE 24
By multiplying both numerator and denominator by the conjugate of the denominator, rationalise the denominator.
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SLIDE 25
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SLIDE 26
By multiplying both numerator and denominator by the conjugate of the denominator, rationalise the denominator.
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SLIDE 27
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SLIDE 28
By multiplying both numerator and denominator by the conjugate of the denominator, rationalise the denominator.
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SLIDE 29
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SLIDE 30
Simplify giving your answer in the form where a and b are integers.
SLIDE 31
SLIDE 32
Write in the form where a and b are integers.
SLIDE 33
SLIDE 34
Example
Exercise
Simplify
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SLIDE 35
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SLIDE 36
Solve each of the following, giving your answers as an exact surd in their simplest form
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SLIDE 37
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SLIDE 38
Solve giving your answers in the form , where a and b are integers.
SLIDE 39
Solve giving your answers in the form , where a and b are integers.
Solution
SLIDE 40
If , which of the following is a possible value of ?
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B
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E
SLIDE 41
If , which of the following is a possible value of ?
Solution
The question asks us to solve for , not for x. Because of this, the question is a �trick� question. In fact, we may directly solve for as follows.
The answer is E
If you solve for x, you obtain
Then , and the same answer.
Note, option C, gives a zero in the denominator of , and is not defined for this value.
SLIDE 42
ERROR IN QUESTION CORRECTED BELOW. CHECK THE SLIDE.
If , find the possible values of
SLIDE 43
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