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www.evamaths.blogspot.in Revision Topic: Powers, Surds and Rational Numbers The objectives of this unit are to: * recap the basic rules for indices; * to evaluate fractional and negative powers of numbers; * to simplify surds and expressions involving surds; * to solve problems involving surds; * to convert a recurring decimal to a fraction. Brief recap of Grade B and C material: Rules of indices: Zero power: Anything to the power 0 is 1 So 70 1 a0 1 ( xy ) 0 1 Multiplying indices: Add the powers So a m a n a mn 83 84 87 52 7 54 73 (52 54 ) (7 73 ) 56 7 4 Dividing indices: Subtract the powers So y m y n y mn 511 54 57 r4 r5 r9 3 3 r6 r r Power of a power: Multiply the powers together So ( a m ) n a mn (93 ) 2 96 Examination style question Simplify each of the following by writing the answer as a single power of 7: a) 74 72 b) 78 73 c) 7 2 6 78 7 2 7 d) 74 1 www.evamaths.blogspot.in Negative powers A power of -1 represents the reciprocal of a number. So: 1 a 1 a 1 7 1 7 You find the reciprocal of a fraction by swapping the numerator and denominator over. Therefore 1 a b b a 1 1 3 3 3 1 1 3 4 4 3 General negative powers are handled as follows: 1 an n a So, 1 1 23 3 2 8 1 1 52 2 5 25 Likewise with fractions: 2 2 2 5 25 (square top and bottom) 5 2 4 3 3 2 3 27 3 2 8 Examination style question Evaluate (i.e. work out) a) 80 b) 32 c) 24 3 5 d) 2 2 www.evamaths.blogspot.in Fractional Powers Fractional powers correspond to roots of numbers: 1 a2 a 1 a3 3 a 1 a4 4 a Therefore 1 42 4 2 1 125 3 3 125 5 1 1 1 1 81 4 1 814 4 81 3 1 27 3 3 27 3 (cube root top and bottom) 1000 1000 10 1 1 1 2 25 2 1 25 2 5 25 1 More complex fractional powers are handled as follows: a m m an n So 4 2 32 5 5 42 5 27 3 9 2 2 3 2 27 3 3 3 9 2 9 3 3 27 4 2 8 4 2 1 1 1 1 1000 3 2 3 2 2 1000 3 ( 1000) 10 100 Examination question Evaluate (i) 52 2 (ii) 83 1 (iii) 49 2 3 www.evamaths.blogspot.in Examination question Find the value of (i) 640 1 (ii) 64 2 2 (iii) 64 3 Surds 6 5 Expressions like 2, 5 3 1 and are all examples of surds as they are expressed in terms of 7 a root. In general, surds are numbers that are left in a form involving a root (typically a square root). Surds are often used when it is important to give an exact answer. For example, suppose we wished to calculate the length of the hypotenuse in this triangle: Using Pythagoras: x cm x 2 42 62 4 cm x 2 16 36 x 2 52 6cm Although we can use a calculator to evaluate the length of the hypotenuse, giving x = 7.21…cm, the decimal would have to be rounded (giving an answer which is not exact). If an exact answer was required, the hypotenuse could be expressed in surd form – the exact length is x = 52 cm. Simplifying Surds Some surds can be simplified. This is done by finding an equivalent expression that involves the square root of a smaller number. The surd n can be simplified if n is divisible by a square number (bigger that 1). To simplify a surd, we use the result: ab a b 4 www.evamaths.blogspot.in Example: Simplify 52 Solution: As 52 is divisible by a square number (4 goes into 52), 52 can be simplified: 52 = 4 13 4 13 2 13 Example 2: Simplify 72 Solution: We know that 72 can be simplified, as 72 can be divided by several square numbers (for example 9 goes into it 8 times). 72 9 8 9 8 3 8 However, this is not the final answer as 8 can also be simplified: 3 8 3 4 2 3 4 2 3 2 2 6 2 So: 72 6 2 Note: This simplified answer could have been obtained directly in one step if 72 had been split up as 36 × 2. Examination style question Simplify 18 Examination style question Simplify 50 Examination style question Simplify 54 5 www.evamaths.blogspot.in Calculating with surds Addition and subtraction of surds The methods involved are similar to those used when simplifying algebraic expressions. Example 1: 4 2 3 3 2 2 3 6 2 4 3 (collect together similar roots) Example 2: 5 7 3 5 2 7 2 5 3 7 5 5 However, it is important to make sure that all surds are simplified as much as possible before trying to collect together like terms. Example 3: Simplify 7 2 3 8 4 18 Solution: Begin by simplifying the surds: 8 4 2 2 2 18 9 2 3 2 Therefore, 7 2 3 8 4 18 = 7 2 3 2 2 4 3 2 7 2 6 2 12 2 2 Example 4: Simplify 5 2 3 3 2 12 50 Solution: First we simplify the surds: 12 4 3 2 3 50 25 2 5 2 So: 5 2 3 3 2 12 50 = 5 2 3 3 2 2 3 5 2 7 3 Worked examination style question 2 72 32 k 2 . Find the value of k. Solution: Simplify the surds: 72 36 2 6 2 32 16 2 4 2 Therefore, 2 72 32 2 6 2 4 2 8 2 So k = 8. Worked examination style question 27 3n . Find the value of n. Solution: 27 9 3 3 3 3 31/ 2 33 / 2 So, n = 3/2 6 www.evamaths.blogspot.in Multiplication of surds We will need to use the following result: a b ab Example: 4 3 2 3 4 2 3 3 = 4 2 3 24 Note: Here we used the fact that 3 3 3 (by definition of the square root of 3). We can think of this as 3 3 9 3 Example 2: 6 2 3 18 6 3 2 18 18 36 18 6 108 Example 3: Expand and simplify: 2 3 32 2 Solution: Use the usual rules for expanding brackets to get: 2 3 3 2 2 2 3 6 2 4 3 6 2 2 3 6 4 Example 4: Expand and simplify: (3 5 2)(3 5 2) Solution: We use any usual method for expanding double brackets: (3 5 2)(3 5 2) = 9 5 5 6 5 6 5 4 = 95 4 = 41 Example 5: Expand and simplify: 5 2 1 2 2 1 Solution: 5 2 1 2 2 1 10 4 5 2 2 2 1 20 3 2 1 19 3 2 7 www.evamaths.blogspot.in Worked examination question Calculate the area of the triangle shown. Solution: 1 5 3 cm Area of triangle = base×height 2 1 = 4 3 5 3 2 = 2 3 5 3 4 3 cm = 10 3 3 = 30 cm2 Examination question 1: 2 Work out the value of 5 3 . Give your answer in the form a b 3 where a and b are integers. Remember: To square a bracket, multiply it by itself. Examination style question : The length of a rectangle is (5 3) cm. The width of the rectangle is (6 3) cm. Work out: a) the perimeter of the rectangle; b) the area of the rectangle. 8 www.evamaths.blogspot.in Examination style question a) Simplify 75 27 b) Simplify 75 27 75 27 Examination question (Edexcel) a) Find the value of n in the equation 2n 8 . A Triangle ABC has an area of 32 cm². b) Calculate the value of k. 8 cm B k C 2 cm 9 www.evamaths.blogspot.in Dividing Surds a a When dividing, we sometimes make use of the result . b b 40 40 Example: 42 10 10 Rationalising a denominator In mathematics, it is considered untidy to leave a surd in the denominator of a fraction. If there is a surd in the denominator, you should try to find an equivalent answer which only has surds on the top of the fraction. This process is called rationalising the denominator. a To rationalise the denominator in , you multiply top and bottom by b: b a a b a b = b b b b Note: Because you multiply the top and the bottom by the same thing, you haven’t changed the value of the number. 6 Example: Rationalise the denominator in . 2 Solution: Multiply top and bottom by 2 : 6 6 2 6 2 = 3 2 2 2 2 2 8 Example: Rationalise the denominator in 3 6 Solution: Multiply top and bottom by 6: 8 8 6 8 6 = 3 6 3 6 6 3 6 8 6 = 18 4 6 = (dividing top and bottom by 2) 9 10 www.evamaths.blogspot.in Examination style question 40 Rationalise the denominator of 5 Examination style question: 12 Express in the form a b where a and b are integers. 3 Examination question 6 a) Express in the form a b where a and b are positive integers. 2 The diagram shows a right-angled isosceles triangle. 6 6 cm The length of each of its equal sides is cm. 2 2 b) Find the area of the triangle. Give your answer as an integer. 6 cm 2 11 www.evamaths.blogspot.in Changing recurring decimals to fractions . . . . . Recurring decimals, such as 0.5 , 0.217 and 0.142 , all can be written as fractions. This is short for the This is short for the This is short for the decimal decimal decimal 0.555555… 0.217217217… 0.142424242…. Note: The dots go over each end of the set of repeating digits. Example 1: . Change 0.5 to a fraction. . Step 1: Write 0.5 as x: x = 0.55555555….. (1) Step 2: Find 10x: 10x = 5.555555555…. (2) Step 3: Subtract (2) – (1): 9x = 5 5 Step 4: Find the fraction: x 9 Example 2: . . a Write 0.217 in the form , where a and b are integers. b . . Step 1: Write 0.217 as x: x = 0.217217217… (1) Step 2: Find 1000x: 1000x = 217.217217… (2) Step 3: Subtract (2) – (1): 999x = 217 217 Step 4: Find the fraction: x 999 Example 3: . . Change 0.142 to a fraction. . . Step 1: Write 0.142 as x: x = 0.142424242… (1) Step 2: Find 100x: 100x = 14.24242424… (2) Step 3: Subtract (2) – (1): 99x = 14.1 14.24242424... 0.14242424... 14.10000000... 14.1 Step 4: Find the fraction: x 99 To change this to a fraction with integers top and bottom, multiply through by 10: 141 x 990 47 This simplifies to 330 12 www.evamaths.blogspot.in Examination question Change to a single fraction . . a) the recurring decimal 0.13 . . b) the recurring decimal 0.513 Examination question 3 a) Change to a decimal. (Hint: divide the numerator by the denominator) 11 . . 13 b) Prove that the recurring decimal 0.39 33 13

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