C1: Surds. Learning objectives

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1 B_Chap0_0-07.qxd 5/6/0 0: am Page CHAPTER C: Surds Learning objectives After studying this chapter, you should be able to: distinguish between rational and irrational numbers understand what is meant by a surd simplify expressions involving surds perform arithmetic involving surds.. Special sets of numbers When you learned to count, you started to use the numbers,,,, which are known as natural numbers and denoted by the symbol. As the need arose to use zero and negative numbers the set of integers, denoted by = {,,, 0,,,,, }, was constructed. Some numbers such as fractions cannot be written as a whole number. Those which can be written in the form a b, where a and b are integers and b is not equal to zero, are called rational numbers and are denoted by, since they can be written as the quotient of two integers. When rational numbers are written as decimals they either terminate such as 0.75 or or they have a sequence of recurring digits such as or Are these sensible or reasonable numbers? The term rational is used because these numbers can be written as the ratio of two integers. However, numbers that cannot be written as a fraction in the form a, where a and b are integers, are called b irrational. The decimal representation of an irrational number neither terminates nor has a recurring pattern of digits, no matter how many decimal places we write down.

2 B_Chap0_0-07.qxd 5/6/0 0: am Page C: Surds Examples include.705 and and.59 which each has a non-repeating pattern of digits in its decimal representation. The set of rationals combined with the set of irrationals gives us the set of real numbers and is denoted by.. Surds In mathematics, we often arrive at answers that contain root signs (they may be square roots, cube roots, etc.). We will find that some of these numbers with a root sign are easy to deal with since they have an exact decimal representation. For instance 6, 8,.56., This is because each of these numbers is rational. Remember that although the equation x 9 has solutions x, the symbol means the positive square root so that 9. Expressions with root signs involving irrational numbers such as 7 or 5 are called surds. EXERCISE A State whether each of the following is a rational or irrational number Sometimes, you will be required to give answers to problems in surd form since these answers are exact. If you use your calculator to get a decimal form, it can only give the answer to a certain number of decimal places and so can only be an approximation. For example, is exact, whereas is the full calculator display, but is still only an approximation to. Order of a surd The order of a surd is determined by the root symbol. For example: is a surd of order (sometimes called a quadratic surd). 5 is a surd of order. n x is a surd of order n. A rational quantity may be expressed in the form of a root of any required order. For example, 9 7 8, etc. In general, you will be handling surds of order.

3 B_Chap0_0-07.qxd 5/6/0 0: am Page 5 C: Surds 5 Worked example. Write the following surds in ascending size: 9, 5, 6 6. It is difficult to see which is the biggest at a glance since they are all of a different order. Since we have the fourth root, the third root and the sixth root of various numbers, we can raise each of the terms to the power, because the lowest common multiple of, and 6 is. Let a 9, b 5 and c 6 6. Hence, a 9 a 9 79 b 5 b 5 65 and c 6 6 c We can now see that b c a and hence rearrange the surds in ascending size as 5, 6 6, 9.. Simplest form of surds When simplifying surds, we try to make the number under the root sign as small as possible. Worked example. Simplify the following as far as possible Hint. Look for a square number that divides exactly into the number under the square root sign (or a perfect cube if the surd is of order, etc.) is a square number that is also a factor of 8. 8 is a perfect cube which is also a factor of but 5 can itself be simplified. is a square number that is a factor of 6.

4 B_Chap0_0-07.qxd 5/6/0 0: am Page 6 6 C: Surds. Manipulating square roots There are some simple rules which apply to all positive numbers and they will help us when working with square roots. The first rule generalises the idea demonstrated in the last worked example. Rules ab a b a b a b Worked example. Simplify the following ( ) EXERCISE B Simplify each of the surd expressions

5 B_Chap0_0-07.qxd 5/6/0 0: am Page 7 C: Surds 7 9 Express each of the following in the form kp where k is an integer and p is a prime number. (a) 7 8, (b) 58 6, (c) Arrange the following surds in ascending size:, 5, Use in geometry One of the main areas in which you will need to work with surds is in the use of Pythagoras theorem in geometry. Often you need the exact value of a particular length. Worked example. The hypotenuse of a right-angled triangle has length 8 cm and one of the other sides has length 6 cm. Find the length of the remaining side. 6 cm 8 cm x cm Let the remaining side have length x cm. By Pythagoras theorem x 6 8 so that x 6 88 x So the remaining side has length cm Like and unlike surds Like surds have the same irrational factor. For example 5, 65, 5 are like surds since they have the same number under the root sign. Unlike surds have different irrational factors. For instance, 7, 6, are unlike surds since they have different numbers under the root sign.

6 B_Chap0_0-07.qxd 5/6/0 0: am Page 8 8 C: Surds Student health warning! It is well worth noting that: a b a b and a b a b For example, consider However, = + = 7. This, of course, is not the same and demonstrates that we cannot split up 9 6 as 9 6. This is a common mistake and must be guarded against..7 Adding and subtracting surds We can add or subtract surds as long as they are like surds. Like surds can be collected. Unlike surds cannot be collected. Worked example.5 Simplify each of the following as much as possible. (a) 5 65, (b) , (c) 96, (d) (a) ; (b) (c) We first need to simplify these surds as far as possible. (5 ) ( 9 ) (5 6 ) These are all like surds and, therefore, can be collected. 9 96; (d) We can collect all the terms with 5 and then separately collect the terms with There are lots of 5 minus 6 lots of 5 giving 8 lots of 5. and 6 are unlike surds and, therefore, cannot be collected so this expression cannot be simplified any further. Note that we cannot combine these any further since 5 and are unlike surds.

7 B_Chap0_0-07.qxd 5/6/0 0: am Page 9 C: Surds 9 EXERCISE C Simplify each of the expressions 0 as far as possible The two shorter sides of a right-angled triangle have lengths m and 6 m. Find the length of the hypotenuse in the form ab metres, where a and b are prime numbers. The hypotenuse of a right-angled triangle has length cm and another one of the sides has length 8 cm. Find the exact length of the third side. Find the length of the remaining side in each of the right-angled triangles below, giving your answers as simply as possible in surd form. (a) cm 5 cm (b) 7 cm 5 cm.8 Multiplying surds We multiply the rational factors then the irrational factors and then simplify (if possible). Worked example.6 Simplify each of the following: You should be able to recognise that and in general n n n.

8 B_Chap0_0-07.qxd 5/6/0 0: am Page 0 0 C: Surds An expression such as 5 is called a compound surd. To multiply compound surds, we use the same idea as multiplying out brackets in algebra. Recall that (a b)(c d) ac bc bd ad. Worked example.7 Simplify each of the following. ( 5)( ). ( )(5 7). ( 5)( 5). ( 5). ( 5)( ) ( )(5 7) ( 5)( 5) ( 5) ( 5)( 5) EXERCISE D Multiply out the following and simplify the answers as far as possible (6 ). 8( 8). 5 5( 65). 6 ( )(7 ). 7 (5 )(5 ). 8 (6 7)(6 7). 9 (0 )(0 ). 0 ( )( ).

9 B_Chap0_0-07.qxd 5/6/0 0: am Page.9 Rationalising the denominator Whenever we have a fraction in which the denominator is a surd, we can rewrite the fraction so that the denominator no longer contains a surd. This is done by multiplying the top and bottom of the fraction by the same number so that the final answer has a rational number in the denominator. Since ( )( ) (), we could simplify by writing ( ). This process is called rationalising the denominator. The method is as follows. If the denominator is of the form a then multiply the top and bottom by a. If the denominator is of the form a b then multiply top and bottom by a b. If the denominator is of the form a b then multiply top and bottom by a b. Since, we could simplify by writing. Since (p q)(p q) p q. C: Surds Worked example.8 Rationalise the denominators of the following: Multiply the top and bottom by ( + 6 ) ( ( 6) ( + 6 ) 9 6) 6 ( 6) We now have a rational number in the denominator. Multiply the top and bottom by ( 6).

10 B_Chap0_0-07.qxd 5/6/0 0: am Page C: Surds. 7 Multiply the top and bottom by ( 7). ( 7 ) 7 7 ( 7) ( 7 ) ( 6 ) ( 5 ) ( 5 ) ( 5 ) 5 9 Note that the number that we multiply the top and bottom by in order to rationalise the denominator is sometimes called the conjugate. The conjugate of 7 is 7. The conjugate of 8 5 is 8 5. Multiply top and bottom by the conjugate (5 ) Although this answer is perfectly correct, it is customary to try to leave a positive denominator, where possible. EXERCISE E Rationalise the denominators of the following:

11 B_Chap0_0-07.qxd 5/6/0 0: am Page C: Surds.0 Equations and inequalities involving surds An equation or inequality may look more complicated because it involves surds. The same techniques are used as in chapter. However, it may be necessary to rationalise the denominator if this is in surd form. Worked example.9 Solve the equation 5 x 5x 7. 5 x 5x x x (5 )x 5 7 x x 5 5 Although this answer is now correct it is better if we rationalise the denominator x 0 9 Worked example.0 Solve the inequality x x 5 x x 5 5 x x 5 ( )x 5 x 5 x Although this answer is again correct, it is better if you rationalise the denominator. In fact, in an examination, you may well be required to do so. 0 0 x x or x EXERCISE F Solve each of the following: x 5 7x. 5x 5 5 x. 7 x 5x x 5 7x.

12 B_Chap0_0-07.qxd 5/6/0 0: am Page C: Surds 5 x x. 6 x 7 5x. 7 x 5x x 5 x x 8 7x. 0 5x x 7. Worked examination question. Express 5 in the form p + q where p and q are rational 7 numbers () 9() which is in the given form p = and q =. We need to rationalise the denominator and do this by multiplying top and bottom by 7. A common mistake is to think that () is 9. This would now score full marks, but it is good to state the actual values of p and q. MIXED EXERCISE Simplify the following as far as possible: (a) 75, (c) 8, (e) 8 98, (b) 00, (d) , (f) Expand the following and simplify where possible. (a) (5 ), (b) 5( 5 ), (c) (50 8 ), (d) ( )( 7), (e) (5 7)(5 ), (f) ( ). Rationalise the denominator, simplifying where possible: (a) 7, (b), 5 (c) 5, (d), 5

13 B_Chap0_0-07.qxd 5/6/0 0: am Page 5 C: Surds 5 (e), (f) 5, 7 (g) 5 7, (h) ( 7). Express each of the following in the form p q where p and q are rational: (a) ( ), (b). ( ) [A] 5 (a) Write down: (i) a rational number which lies between and 5, (ii) an irrational number which lies between and 5. (b) A student says, When you multiply two irrational numbers together the answer is always an irrational number. Simplify ( )( ) and comment on the student s statement. [A] 6 Express each of the following in the form p q7 where p and q are rational numbers: (a) ( 7)(5 7), (b) ( 5 7 ) ( 7 ). [A] 7 (a) Express each of the following in the form k5: (i) 5, (ii) (b) Hence write 5 5 in the form n5, where n is an integer. [A] 8 (a) Express (7 ) in the form a b7, where a and b are integers. (7 ) (b) Hence express in the form p q7, where (7 ) p and q are rational numbers. [A] 9 Express each of the following in the form p q: (a) ( )(5 ), 6 (b). [A] 0 (a) Express in the form a b, where a and b are integers. (b) Solve the inequality (x ) x. [A]

14 B_Chap0_0-07.qxd 5/6/0 0: am Page 6 6 C: Surds Key point summary A rational number is one which can be written in the p form a, where a and b are integers. b A real number that is not rational is called irrational. p Its decimal representation neither terminates nor has a recurring pattern of digits. A surd is an irrational number containing a root sign. p ab a b 5 a b a b p6 p6 6 Like surds can be collected. p8 Unlike surds cannot be collected. 7 To rationalise the denominator of the form a, p multiply top and bottom by a. 8 To rationalise the denominator of the form a b, p multiply top and bottom by a b. 9 To rationalise the denominator of the form a b, p multiply top and bottom by a b. Test yourself State whether each of the following is rational or irrational: Section. (a) 8, (b), (c), (d). 5 Simplify each of the surd expressions below: Sections. and. (a) 7 6, (b) 9 0, (c) Simplify each of the following surd expressions as far Section.7 as possible. (a) 7, (b) 0 5, (c) 7. Find the length of the hypotenuse of a right-angled triangle Section.5 if the two smaller sides have lengths cm and 5 cm. 5 Rationalise the denominator of the expression: Section What to review

15 B_Chap0_0-07.qxd 5/6/0 0: am Page 7 C: Surds 7 Test yourself ANSWERS (a) rational; (b) irrational; (c) irrational; (d) rational. (a) 9; (b) 6; (c) 5. (a) ; (b) 55; (c) cannot simplify. 6 cm