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## Maths Age 14-16

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**Maths Age 14-16**N2 Powers, roots and standard form**N2 Powers, roots and standard form**Contents • A N2.2 Index laws • A N2.3 Negative indices and reciprocals • A N2.1 Powers and roots N2.4 Fractional indices • A N2.5 Surds • A N2.6 Standard form • A**Square numbers**When we multiply a number by itself we say that we are squaring the number. To square a number we can write a small 2 after it. For example, the number 3 multiplied by itself can be written as 3 × 3 or 32 The value of three squared is 9. The result of any whole number multiplied by itself is called a square number.**Square roots**squared square rooted 64 = Finding the square root is the inverse of finding the square: 8 64 We write 8 The square root of 64 is 8.**The product of two square numbers**The product of two square numbers is always another square number. For example, 4 × 25 = 100 because 2 × 2 × 5 × 5 = 2 × 5 × 2 × 5 and (2 × 5)2 = 102 We can use this fact to help us find the square roots of larger square numbers.**Using factors to find square roots**Find 400 Find 225 If a number has factors that are square numbers then we can use these factors to find the square root. For example, √400 = √(4 × 100) √225 = √(9 × 25) = √4 × √100 = √9 × √25 = 2 × 10 = 3 × 5 = 20 = 15**Finding square roots of decimals**Find 0.09 Find 0.0144 We can also find the square rootof a number can be made be dividing two square numbers. For example, 0.09 = (9 ÷ 100) 0.0144 = (144 ÷ 10000) = √9 ÷√100 = √144 ÷√10000 = 3 ÷ 10 = 12 ÷ 100 = 0.3 = 0.12**Approximate square roots**key on your calculator to find out 2. Use the If a number cannot be written as a product or quotient of two square numbers then its square root cannot be found exactly. The calculator shows this as 1.414213562 This is an approximation to 9 decimal places. The number of digits after the decimal point is infinite and non-repeating. This is an example of an irrational number.**Estimating square roots**What is 50? Use the key on you calculator to work out the answer. 50 is not a square number but lies between 49 and 64. 50 is much closer to 49 than to 64, so 50 will be about 7.1 Therefore, 49 < 50 < 64 So, 7 < 50 < 8 50 = 7.07 (to 2 decimal places.)**Negative square roots**5 × 5 = 25 and –5 ×–5 = 25 Therefore, the square root of 25 is 5 or –5. When we use the symbol we usually mean the positive square root. We can also write ± to mean both the positive and the negative square root. However the equation, x2 = 25 has 2 solutions, or x = 5 x = –5**Cubes**The numbers 1, 8, 27, 64, and 125 are all: Cube numbers 13 = 1 × 1 × 1 = 1 ‘1 cubed’ or ‘1 to the power of 3’ 23 = 2 × 2 × 2 = 8 ‘2 cubed’ or ‘2 to the power of 3’ 33 = 3 × 3 × 3 = 27 ‘3 cubed’ or ‘3 to the power of 3’ 43 = 4 × 4 × 4 = 64 ‘4 cubed’ or ‘4 to the power of 3’ 53 = 5 × 5 × 5 = 125 ‘5 cubed’ or ‘5 to the power of 3’**Cube roots**cubed cube rooted 125 = 5 3 Finding the cube root is the inverse of finding the cube: 5 125 We write The cube root of 125 is 5.**We use index notation to show repeated multiplication by the**same number. Index notation For example: we can use index notation to write 2 × 2 × 2 × 2 × 2 as Index or power 25 base This number is read as ‘two to the power of five’. 25 = 2 × 2 × 2 × 2 × 2 = 32**Evaluate the following:**Index notation 62 = 6 × 6 = 36 When we raise a negative number to an odd power the answer is negative. 34 = 3 × 3 × 3 × 3 = 81 (–5)3 = –5 × –5 × –5 = –125 27 = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 128 When we raise a negative number to an even power the answer is positive. (–1)5 = –1 × –1 × –1 × –1 × –1 = –1 (–4)4 = –4× –4 × –4 × –4 = 256**7**4 = xy Using a calculator to find powers We can use the xykey on a calculator to find powers. For example: to calculate the value of 74 we key in: The calculator shows this as 2401. 74 = 7 × 7 × 7 × 7 = 2401**N2 Powers, roots and standard form**Contents N2.1 Powers and roots • A • A N2.3 Negative indices and reciprocals • A N2.2 Index laws N2.4 Fractional indices • A N2.5 Surds • A N2.6 Standard form • A**When we multiply two numbers written in index form and with**the same base we can see an interesting result. Multiplying numbers in index form For example: 34 × 32 = (3 × 3 × 3 × 3) × (3 × 3) = 3 × 3 × 3 × 3 × 3 × 3 = 36 = 3(4 + 2) 73 × 75 = (7 × 7 × 7) × (7 × 7 × 7 × 7 × 7) = 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 = 78 = 7(3 + 5) When we multiply two numbers with the same base the indices are added. In general, xm × xn = x(m + n) What do you notice?**4 × 4 × 4 × 4 × 4**= 4 × 4 5 × 5 × 5 × 5 × 5 × 5 = 5 × 5 × 5 × 5 When we divide two numbers written in index form and with the same base we can see another interesting result. Dividing numbers in index form For example: 45 ÷ 42 = 4 × 4 × 4 = 43 = 4(5 – 2) 56 ÷ 54 = 5 × 5 = 52 = 5(6 – 4) When we divide two numbers with the same base the indices are subtracted. In general, xm÷xn=x(m – n) What do you notice?**Raising a power to a power**Sometimes numbers can be raised to a power and the result raised to another power. For example, (43)2 = 43 × 43 = (4 × 4× 4) × (4 × 4× 4) = 46 = 4(3 ×2) When a number is raised to a power and then raised to another power, the powers are multiplied. In general, (xm)n=xmn What do you notice?**61**471 0.91 –51 01 The power of 1 Find the value of the following using your calculator: Any number raised to the power of 1 is equal to the number itself. In general, x1 = x Because of this we don’t usually write the power when a number is raised to the power of 1.**That means that:**60 = 1 Look at the following division: The power of 0 64 ÷ 64 = 1 Using the second index law, 64 ÷ 64 = 6(4 – 4) = 60 Anynon-zero number raised to the power of 0 is equal to 1. For example, 100 = 1 3.4520 = 1 723 538 5920 = 1**x0=1 (for x = 0)**Here is a summery of the index laws you have met so far: Index laws xm × xn = x(m + n) xm÷xn=x(m – n) (xm)n=xmn x1=x**N2 Powers, roots and standard form**Contents N2.1 Powers and roots • A N2.2 Index laws • A • A N2.3 Negative indices and reciprocals N2.4 Fractional indices • A N2.5 Surds • A N2.6 Standard form • A**=**= 1 3 × 3 1 1 1 1 3 × 3 × 3 × 3 3 × 3 32 32 74 53 1 6 Look at the following division: Negative indices 32 ÷ 34 = Using the second index law, 32 ÷ 34 = 3(2 – 4) = 3–2 That means that 3–2 = Similarly, 6–1 = 7–4 = and 5–3 =**Reciprocals**1 a b The reciprocal of a is a b a The reciprocal of is x-1 We can find reciprocals on a calculator using the key. A number raised to the power of –1 gives us the reciprocalof that number. The reciprocal of a number is what we multiply the number by to get 1.**Finding the reciprocals**1 1 3 5 4 4 7 7 3 3 or The reciprocal of 6 = 6-1 = 3 6 6 7 7 3 4 5 7 5 –1 2) The reciprocal of = The reciprocal of = = or 0.8 = Find the reciprocals of the following: 1) 6 3) 0.8 = 1.25 or 0.8–1 = 1.25**1**x–1 = x 1 1 The reciprocal of x is The reciprocal of xn is x xn 1 x–n = xn Index laws for negative indices Here is a summery of the index laws for negative indices.**N2 Powers, roots and standard form**Contents N2.1 Powers and roots • A N2.2 Index laws • A N2.3 Negative indices and reciprocals • A N2.4 Fractional indices • A N2.5 Surds • A N2.6 Standard form • A**1**1 1 1 1 1 1 1 1 1 1 1 1 9×9= 9+= 2 2 2 2 3 3 2 2 3 3 3 3 3 x = x 8 + + = 8×8 ×8 = 8 × 8 × 8 = 8 3 3 3 x = x 3 Indices can also be fractional. Suppose we have 9. Fractional indices 91 = 9 Because 3 × 3 = 9 But, 9 × 9 = 9 In general, 81 = 8 Similarly, Because 2 × 2 × 2 = 8 But, In general,**3**1 3 1 What is the value of 25 ? 2 2 2 2 m n 25 We can think of as 25 . × 3 25 × 3 = (25)3 n x = (x)m Fractional indices Using the rule that (xa)b=xab we can write = (5)3 = 125 In general,**Evaluate the following**2 1 1 2 5 1 5 1 2 2 1 2 1) 49 49 = 3 3 3 3 2 3 3 2 3 3 2 2 2) 1000 1000 = 1 1 1 3) 8- 8- = = = 2 3√8 8 1 1 1 1 4) 64- = = = = 64- 64 (3√64)2 42 16 5) 4 4 = √49 = 7 (3√1000)2 = 102 = 100 (√4)5 = 25 = 32**1**2 x = x 1 m n n x = x n x = xm or (x)m n n Here is a summery of the index laws for fractional indices. Index laws for fractional indices**N2 Powers, roots and standard form**Contents N2.1 Powers and roots • A N2.2 Index laws • A N2.3 Negative indices and reciprocals • A N2.5 Surds N2.4 Fractional indices • A • A N2.6 Standard form • A**Surds**The square roots of many numbers cannot be found exactly. For example, the value of √3 cannot be written exactly as a fraction or a decimal. The value of √3 is an irrational number. For this reason it is often better to leave the square root sign in and write the number as √3. √3 is an example of a surd. Which one of the following is not a surd? √2, √6, √9or √14 9 is not a surd because it can be written exactly.**Multiplying surds**What is the value of √3 × √3? We can think of this as squaring the square root of three. Squaring and square rooting are inverse operations so, √3 × √3 = 3 In general, √a × √a = a What is the value of √3 × √3 × √3? Like algebra, we do not use the × sign when writing surds. Using the above result, √3 × √3 × √3 = 3 × √3 = 3√3**Multiplying surds**Use a calculator to find the value of √2 × √8. What do you notice? √2 × √8 = 4 (= √16) 4 is the square root of 16 and 2 × 8 = 16. Use a calculator to find the value of √3 × √12. √3 × √12 = 6 (= √36) 6 is the square root of 36 and 3 × 12 = 36. In general, √a × √b = √ab**Dividing surds**a b In general, √a÷ √b = Use a calculator to find the value of √20 ÷ √5. What do you notice? √20 ÷ √5 = 2 (= √4) 2 is the square root of 4 and 20 ÷ 5 = 4. Use a calculator to find the value of √18÷ √2. √18÷ √2 = 3 (= √9) 3 is the square root of 9 and 18 ÷ 2 = 9.**Simplifying surds**We are often required to simplify surds by writing them in the form a√b. For example, Simplify √50 by writing it in the forma√b. Start by finding the largest square number that divides into 50. This is 25. We can use this to write: √50 = √(25 × 2) = √25 × √2 = 5√2**Simplifying surds**Simplify the following surds by writing them in the forma√b. 1) √45 2) √24 3) √300 √45 = √(9 × 5) √24 = √(4 × 6) √300 = √(100 × 3) = √9 × √5 = √4 × √6 = √100 × √3 = 3√5 = 2√6 = 10√3**Adding and subtracting surds**Surds can be added or subtracted if the number under the square root sign is the same. For example, Simplify √27 + √75. Start by writing √27 and √75 in their simplest forms. √27 = √(9 × 3) √75 = √(25 × 3) = √9 × √3 = √25 × √3 = 3√3 = 5√3 √27 + √75 = 3√3 + 5√3 = 8√3**Perimeter and area problem**6 1 2 3 The following rectangle has been drawn on a square grid. Use Pythagoras’ theorem to find the length and width of the rectangle and hence find its perimeter and area in surd form. Width = √(32 + 12) = √(9 + 1) 2√10 = √10 units √10 Length = √(62 + 22) = √(36 + 4) = √40 = 2√10 units**Perimeter and area problem**The following rectangle has been drawn on a square grid. Use Pythagoras’ theorem to find the length and width of the rectangle and hence find its perimeter and area in surd form. Perimeter = √10 + 2√10 + 6 1 √10 + 2√10 2√10 2 3 = 6√10 units √10 Area = √10 ×2√10 = 2 × √10 × √10 = 2 × 10 = 20 units2**Rationalizing the denominator**Simplify the fraction 5 √2 When a fraction has a surd as a denominator we usually rewrite it so that the denominator is a rational number. This is called rationalizing the denominator. Remember, if we multiply the numerator and the denominator of a fraction by the same number the value of the fraction remains unchanged. In this example, we can multiply the numerator and the denominator by √2 to make the denominator into a whole number.**Rationalizing the denominator**Simplify the fraction ×√2 5 5 5√2 √2 = √2 ×√2 When a fraction has a surd as a denominator we usually rewrite it so that the denominator is a rational number. This is called rationalizing the denominator. 2**Rationalizing the denominator**√2 3 1) 2) 3) √5 4√7 ×√3 ×√5 ×√7 2 2√3 √2 √10 3 3√7 = = = 2 √3 √5 4√7 √3 ×√3 ×√5 ×√7 Simplify the following fractions by rationalizing their denominators. 3 5 28