Simplifying Surds Slideshow 6, Mathematics, Mr. Richard Sasaki Objectives Understand the meaning of rational numbers Understand the meaning of surd Be able to check whether a number is a surd or not Be able to simplify surds Rational Numbers What is a rational number? A rational number is a number that can be written in the form of a fraction where the numerator and denominator are integers. is rational if where . If where , we say . ( is in the rational number set, .)
Rationality If a number is not rational, we say that it is irrational . is irrational if it cant be written in the form where . Therefore, an irrational number . Example Show that 0.8 . . 4 As . 5 Note: If , (You cant divide a number by .) and . .
=1 =5 =2 , 4 16=2 4=8 They are the same. . 14 6 6 9 =13( 120 3 13=9) = 28( 120 3 28=6 )
=37( 120 3 37=3) 54 12 Surds What is a surd? A surd is an irrational root of an integer. We cant remove its root symbol by simplifying it. Are the following surds? 2 9 Yes! No! 2 5 32
Yes! Yes! Even if the expression is not fully simplified, if it is a root and irrational, it is a surd. Multiplying Roots How do we multiply surds and roots? Lets consider two roots, and where . If we square both sides, we get 2 2 =( ) 2 =
2 2 2 =( ) ( ) 2 =
If we square root both sides, we get = , where . Simplifying Surds To simplify a surd, we need to write it in the form where is as small as possible and . Note: Obviously, if . Example Simplify . 8= 4 2= 4 2=2 2 We try to take remove square factors out and simplify them by removing their square root symbol. 4 2
4 6 2 37 Yes 16 2 8 21 6 No, for example, is a surd but 6 is not prime. Square numbers. Hard Answers (Questions 1 4) It must be a number with no square factors. (For example , we cant divide this by anything square apart from .) 8 3
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Irrational Numbers don't repeat in a pattern, never terminate, and cannot be expressed as a ratio of two integers. Examples of Rational Numbers: 8/9, 2.66666666, 3.0. ... When we add or subtract polynomials we DO NOT mess with the exponents!
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