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Name of Lecturer: Mr. J.Agius Course: HVAC1 Lesson 25 Chapter 5: Indices and Standard form Square roots and Cube Roots Square Root The square root of a given number is a number that, when squared, exactly equals the given number. The square root or radical sign is . 64 8 because 82 = 64 0.09 0.3 because (0.3)2 = 0.09 Taking a root is the inverse operation of raising to a power in the same way that subtraction is the inverse of addition and division is the inverse of multiplication. Numbers whose square roots are exactly whole numbers, fractions, or finite decimals, such as 64 and 0.09 shown above, are called perfect squares. The perfect squares of the whole numbers 1 to 12 are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 You should learn these perfect squares to help you understand the ideas better and do calculations with square roots more effectively. Example 1 Find the square roots without the calculator: a) 16 49 121 144 b) Answer To find the square root of a fraction, find the square root of the numerator and the denominator separately: 2 a) 16 16 4 4 2 16 4 because 2 49 49 7 49 7 7 b) 121 121 11 112 121 11 because 2 144 144 12 144 12 12 2 5 Indices and Standard form Page 1 Name of Lecturer: Mr. J.Agius Course: HVAC1 Example 2 Find the square roots without the calculator: a) 1.44 b) 0.0025 Answer a) When you square a decimal that is a perfect square the number of decimal places doubles. Therefore, the square root of a decimal will have half as many decimal places as the original number. Since 1.44 has two decimal places, its square root has one decimal place. Knowing that 122 = 144, it follows that 1.22 = 1.44 and therefore: 1.44 1.2 b) The square root of a number less than one is greater than the number itself. This is true because, when you square a number less than one, the result is less than the original number. For example, 0.32 = 0.09. Since the number in the square root, 0.0025, has four decimal places, its square root has two decimal places. Knowing that 52 = 25, it follows that 0.052 = 0.0025 and therefore: 0.0025 0.05 Numbers that are not perfect squares, such as 2, 3, 5, 7, etc. have square roots that are infinite decimals, and it is necessary to round off the calculator result: 2 1.414213562... 1.414 7 2.645751311... 2.646 The symbol “” means approximately equal. That is, 2 is approximately equal to 1.414, and 7 is approximately equal to 2.646. When square roots are infinite decimals, it is sometimes easier to calculate with them in radical form instead of as decimals. The following definition of the square root for any number x is useful in such calculations: Definition of Square Root x2 x 2 x (3.1) The definition says that the operations of square and square root are inverse operations and “cancel each other out.” When a square root and a square appear together, the radical and the square can be eliminated. For example: 5 Indices and Standard form Page 2 Name of Lecturer: Mr. J.Agius Course: HVAC1 9 2 9 and 9 2 9 2 2 22 2 2 and 2 5 5 5 5 Two basic rules for roots of positive numbers x and y are: Rules for Roots xy x y x y (3.2) x (3.3) y Rules (3.2) and (3.3) are used to simplify radicals. You can apply them working from left to right, or from right to left. Products (or quotients) under a radical sign can be separated into products (or quotients) of separate radicals and vice versa. Example 3 Find the square roots without the calculator: a) 14400 b) 324 Answer Use rule (3.2) to find the square root of a large perfect square by factorising the number into small perfect squares. Apply the rule from left to right, separating the radical into the product of two radicals containing perfect squares: 144100 144 100 1210 120 481 4 81 29 18 (a) 14400 (b) 324 Example 4 Simplify the product a) 0.2 1.8 b) 0.64 0.0256 c) 28 7 Answer Apply rule (3.2) from right to left and multiply under one radical: a) b) c) 0.2 1.8 0.21.8 0.64 0.64 0.0256 0.0256 28 7 0.36 0.6 0.8 40.0064 0.8 4 0.0064 0.8 0.80 5 20.08 0.16 28 42 7 5 Indices and Standard form Page 3 Name of Lecturer: Mr. J.Agius Course: HVAC1 Cube Roots A cube root is the inverse of raising to the third power. A cube root is written using the radical sign with the index 3 in the crook of the radical sign (for a square root the index 2 is understood): 1000 10 because 103 = 1000 0.064 0.4 because 0.43 = 0.064 3 3 The first six perfect cubes are: 3 8 2, 3 27 3 , 3 64 4 , 3 125 5 , 3 216 6 It is helpful to know these perfect cubes. Rules (3.2) and (3.3) for square roots also work for cube roots, as shown in the next example. Example 5 Find the cube root of 3 8 125 Answer Apply rule 3.3 for cube roots. Separate the fraction into two cube roots: 3 3 8 8 2 3 125 125 5 The definition of a cube root for a number x is similar to that for a square root: Definition of Cube Root 3 x3 x 3 3 x (3.4) Since raising to the third power is the inverse of taking a cube root, the two operations cancel each other. For example: 3 83 8 and 8 3 3 8 The cube root of a number can be found on the calculator by using the cube root key. Note also that 3 1 3 x x . This rule is valid to any root. Raising to Positive Powers with Powers of Ten Raise the number to the positive power and multiply the exponents to obtain the power of ten. (5 102)3 = 53 10(2)(3) = 125 106 5 Indices and Standard form Page 4 Name of Lecturer: Mr. J.Agius Course: HVAC1 Example 6 a) (1.5 10)2 Perform the operations b) (1.84 103)3 Answer a) Square the 1.5 and multiply the power of ten by 2: (1.5 10)2 = 1.52 101(2) = 2.25 102 b) Raise 1.84 to the third power and multiply the power of 10 by 3. (1.84 103)3 = 1.843 103(3) 6.23 109 Taking Roots with Powers of Ten Take the root of the number and divide the root index into the power of ten. 6 2 81 10 81 10 9 10 3 6 Example 7 Simplify a) 2.25 10 4 b) 27 10 9 3 Answer a) Take the square root of 2.25 and divide the exponent 4 by the index 2: 4 2 2.25 10 2.25 10 1.5 10 2 4 b) Take the cube root of 27 and divide the exponent 9 by the index 3. 3 9 3 27 10 27 10 3 10 3 9 3 Example 8 Calculate and give the answer in power of ten, and in ordinary notation: 9 10 6 2 10 3 2 Answer Apply the order of operations, and do the powers and roots first. Find the square root of the numerator and raise the number to the power in the denominator. 9 10 6 2 10 3 2 Then do the division: 5 Indices and Standard form 6 9 10 2 3 10 3 2 2 10 32 4 10 6 3 10 3 3 10 36 0.75 10 3 0.00075 4 10 6 4 Page 5 Name of Lecturer: Mr. J.Agius Course: HVAC1 Square roots and Cube roots Q1 Work out the following without using the calculator. Do each exercise by hand and give the answers in decimal or fraction form. a) 16 b) e) 16 36 f) i) 0.01 j) 121 4 81 2.25 m) 3 27 n) 3 125 q) 3 0.001 r) 3 0.008 Q2 c) 25 4 d) 9 100 g) 0.64 h) 0.36 k) 0.0081 l) 0.0144 1 8 p) o) 3 3 64 1000 Simplify each expression by applying the rules for radicals. Do each exercise by hand and give the answers in decimal or fraction form. a) 10 2 b) 0.12 c) 784 d) 576 e) 1.96 f) 4.84 g) 8 18 h) 12 27 i) 0.4 0.9 j) 0.02 0.08 k) n) 0 .2 0 .8 m) Q3 a) e) Q4 0.09 9 12 3 (4 103)3 (0.8 105)2 b) (6 102)2 f) (0.1 10)2 c) (2 102)4 g) (1 103)(3 104)2 (3 104)3 (4 10)(2 103)3 d) h) Work out the following without the use of a calculator and express the answer in terms of power of ten. 16 10 6 b) 25 10 4 c) e) 0.25 10 4 f) 0.64 10 6 g) 3 k) m) 75 Work out the following without the use of a calculator and express the answer in terms of power of ten. a) i) 3 l) 9 10 4 10 4 2 36 10 2 2 10 3 2 5 Indices and Standard form j) n) 110 25 10 6 64 10 4 5 10 2 2 2 144 10 2 d) 8 10 9 h) 9 10 8 15 10 3 l) 3 10 3 2 o) 144 10 2 49 1012 3 27 10 3 16 10 2 8 10 2 6 10 2 2 p) 81 10 4 Page 6 Name of Lecturer: Mr. J.Agius Course: HVAC1 Applied Problems Q4 In the following problems try to solve each by hand. Check with the calculator. a) The area of a square plot of land is A = 4900 ft2. Find the side of the square s b) The area A of a square circuit board is 10,000 mm2 (square millimetres). How long is the side s A of the circuit board in mm and cm (centimetres)? c) Find the hypotenuse of a right triangle given by: c 1.5 2 2.0 2 d) The radius of a sphere is given approximately by: r 3 A. V where V = volume. Find r in ft, 4.2 when V = 0.0042 ft3. Applications to Electronics Q5 Work out with the use of a calculator a) The impedance of an alternating current (ac) circuit is given by: Z 16 2 12 2 Impedance is measured in ohms like resistance. Find the value of Z in ohms (). b) As in problem (a), find the impedance Z in ohms of an ac circuit given by: Z 15 2 20 2 c) Find the voltage V in volts (V) in a circuit given by: V 12030 d) Find the current I in amps (A) in a circuit given by: I 4 10000 e) Find the current I in amps (A) in a circuit given by: I 10 150 100 f) Find the voltage V in volts (V) in a circuit given by: V 16100 44 3.2 10 P 3 2 g) Find the power P in watts (W) dissipated as heat in a resistor given by: h) Find the power P in watts (W) dissipated as heat in a resistor given by: P = (0.12)2(0.5 103) i) Find the current I in amps (A) in a series circuit given by: I 1.6 10 6 24 5.7 10 3.9 10 3 3 Change to ordinary notation before calculating. j) Find V the voltage 166.110 3 drop 3.9 10 5 Indices and Standard form 3 V in volts across two resistances given by: Page 7