Y9 Maths Indices, Roots and Rational Numbers PDF

# Indices, Roots and Rational Numbers ## You Will Learn How To: * Use positive, negative and zero indices, and the index laws for multiplication and division. * Understand the difference between rational and irrational numbers. * Use knowledge of square and cube roots to estimate surds. ## Starti...

# Indices, Roots and Rational Numbers ## You Will Learn How To: * Use positive, negative and zero indices, and the index laws for multiplication and division. * Understand the difference between rational and irrational numbers. * Use knowledge of square and cube roots to estimate surds. ## Starting Point Do you remember... * the meanings of positive and zero indices? For example, write the values of 3², 2⁵, and 7⁰. * the index laws for multiplication and division? For example, write each expression as a single power of 9: 9³ x 9⁵, 9⁷ ÷ 9⁴ * the definition of a rational number? For example, find all of the rational numbers in this list: -7, ¹²/₁₃, π, 0.3, 5.97 * the values of squares and cubes of the first few integers? For example, write the square root of 81 and the cube root of 125. ## This Will Also Be Helpful When... * you learn the power of a power rule for indices * you learn to simplify and manipulate surds. ## 1.0 Getting Started The table shows powers of 2. It includes negative indices. | | | | ------ | --------------------- | | 2⁴ = 2 x 2 x 2 x 2 = 16 | ÷ | | 2³ = | ÷ | | 2² = | ÷ | | 2¹ = | ÷ | | 2⁰ = | ÷ | | 2^-1 = | ÷ | | 2^-2 = | ÷ | | 2^-3 = | ÷ | | 2^-4 = | ÷ | Copy the table and complete the rows for the positive and zero indices. Now complete the labels on the arrows. Can you use the pattern to complete the rows for the negative indices? Describe what a negative index means. Make a similar table for powers of 5. ## 1.1 Indices ### Key Terms * A **positive power** or **index** tells you how many **copies** of a number to multiply together. For example, 2⁵ means 2 x 2 x 2 x 2 x 2 = 32. * Any number to the **power** of 0 **equals** 1. * A **negative power** or **index** tells you how many **copies** of a number to **divide** by. For example, 2^-3 means 1 ÷ 2 ÷ 2 ÷ 2 or ¹/_{2 x 2 x 2}, which **equals** ¹/₈. This is the **reciprocal** of 2³. * To **multiply powers** of a number, **add** the powers. For example, 3³ x 3² = 3³⁺² = 3⁵. * To **divide powers** of a number, **subtract** the powers. For example, 3³ ÷ 3² = 3^3-2 = 3¹ = 3. * A **base** is a number that has an **index** attached. For example, in 2⁵, the **base** is 2. ### Did You Know? There is a unit of length called a **nanometre**, which is 10^-9 m or 0.000000001 m. Viruses measure about 20 to 400 nanometres across. They can only be seen with very powerful microscopes. ## Worked Example 1 **Write the value of 2^-5** 2^-5 = ¹/_2⁵ = ¹/_{2 x 2 x 2 x 2 x 2} = ¹/₃₂ 2^-5 is the **reciprocal** of 2⁵ 2⁵ = 32, so the **reciprocal** of 2⁵ is ¹/₃₂ ## Worked Example 2 **Write each expression as a power of 4.** a) 4² x 4^-3 a) 4² x 4^-3 = 4^2-3 = 4^-1 b) 4^-2 ÷ 4^-4 b) 4^-2 ÷ 4^-4 = 4^-2-(-4) = 4² c) (4^-1)³ To **multiply** two powers of the same number, **add** the indices. To **divide** a power by another **power** of the same **number**, subtract the second index from the first index. To **cube** a number, multiply three copies of the number together. **Add** the indices. c) (4^-1)³ = 4^-1 x 4^-1 x 4^-1 = 4^-1-1-1 = 4^-3 ## Exercise 1 1. This table shows some powers of 2. Copy and complete the table. Describe the pattern. | Power | Value | | :---- | :---- | | 2^-4 | | | 2^-3 | | | 2^-2 | | | 2^-1 | | | 2⁰ | | | 2¹ | | | 2² | | | 2³ | | | 2⁴ | | 2. Which of these numbers are less than 0? 5⁰, (-2)^-2, 2^-3, (-3)^-1, -4^-2, -5² 3. Write each number as a power of 3. a) 9 b) ¹/₃ c) 81 d) ¹/₉ e) 27 f) 1 g) 243 h) ¹/₈₁ ## Exercise 2 1. Here is a list of numbers: -7.2, √8, -21, 13, ¹/₁₁, -√4 Write down a number that is: a) both rational and an integer b) rational but not an integer c) irrational. 2. Here is a list of numbers: -5, 22, 7, √12, 14.75, √81, 0.3, π Sort the numbers into rational and irrational numbers. 3. Sort these numbers into rational and irrational numbers: √6, √4, √5, √8, √1, -√27 4. Give an example where: a) the product of two irrational numbers is a rational number b) the product of two rational numbers is a rational number c) the product of a rational number and an irrational number is an irrational number. 5. Decide whether each statement is true or false. If it is true, give an example. a) An integer can be an irrational number. b) An irrational number can be negative. c) The square root of a natural number can be a natural number. d) The square root of a natural number can be a negative integer. e) The square root of a natural number can be a rational number. f) The square root of a natural number can be an irrational number. ## Exercise 3 1. Write down the value of : a) √81 b) √144 c) √256 d) √1 e) √64 f) √125 2. a) Copy and complete this number line. [image of number line] b) Use the number line to estimate a value for: i) √40 ii) √70 iii) √93 iv) √35 v) √54 vi) √100 3. Is each statement true or false? a) 6 < √44 < 7 b) 9 < √104 < 10 c) 11 < √120 < 12 d) 11 < √125 < 12 e) 2 < √5 < 3 f) 3 < √11 < 4 g) 5 < √29 < 6 h) 3 < √45 < 4 4. Estimate each value to one decimal place. a) √30 b) √68 c) √117 d) √122 e) √10 f) √24 g) √70 h) √113 5. [square root] is an integer. Find all of the possible values of [square root] if : a) 1 < [square root] < 2 b) 2 < [square root] < 3 c) 1 < [square root] < 2 6. Points A, B, C, D, and E on the number line below correspond to one number from the list below: √2, √1, √5, √30, √23 [image of number line] Match each number above with one of the points on the number line. 7. Is each statement true or false? a) √19 < √60 b) √100 > √8 c) 8.2² < 64 d) √80 < 2.7³ ## 1.2 Rational and Irrational Numbers ### Key Terms * The **natural numbers** are the **positive integers**: 1, 2, 3, 4, 5, ... * A **rational number** is a number that can be written as a **fraction**. * An **irrational number** is a number that cannot be written as a **fraction**. ### Did You Know? The **square root** of a **negative number** is called an **imaginary number**. The square root of negative one, √-1, has the symbol i in mathematics. Imaginary numbers are different from **real numbers**. They are very useful for doing calculations in some areas of science, such as **electrical engineering** and **quantum physics**. ## Worked Example 3 Here is a list of numbers: √18, 0.16, 0.867, -5, √6, ¹⁹/₇, 6^-7, π, √-4, √8, √-8 Write down the numbers that are: a) rational b) irrational c) neither rational nor irrational. a) **rational** * 0.16 All recurring decimals can be written as fractions: 0.16 = ¹/₆ * 0.867 Any terminating decimal can be written as a fraction: 0.867 = ⁸⁶⁷/₁₀₀₀ * -5 Any integer can be written as a fraction: -5 = ^-5/₁ * ¹⁹/₇ Any mixed number can be written as an improper fraction: 1 ²/₇ = ¹³/₇ * √8 = ³/₈ = 2, which is rational. * √-8 = -2, which is rational. b) **irrational** * √18 The square root of any number that is not a square number is irrational. * √6 The cube root of any number that is not a cube number is irrational. * π cannot be written exactly as a fraction. It is a decimal that neither terminates nor recurs. c) **neither rational nor irrational.** * √-4 You will not be expected to work with numbers like these. ## 1.3 Estimating Surds ### Key Terms A **surd** is a square root of a number that is not a square number, or a cube root of a number that is not a cube number. For example √2, √12, and √15 are surds, but √4 and ³√27 are not surds. Surds are irrational. ## Worked Example 4 **Estimate: ** a) √55 b) ³√20 a) 7² = 7 x 7 = 49 8² = 8 x 8 = 64 49 < 55 < 64 √49 < √55 < √64 7 < √55 < 8 [image of number line] 55 is between 7² = 49 and 8² = 64. 55 is closer to 49 than to 64, so √55 is closer to 7 than to 8. √55 is about 7.4 (or 7.5) b) 2³ = 8 3³ = 27 8 < 20 < 27 ³√8 < ³√20 < ³√27 2 < ³√20 < 3 [image of number line] 20 is between 2³ = 8 and 3³ = 27. 20 is closer to 27 than to 8, so ³√20 is closer to 3 than to 2. ³√20 is about 2.7 (or 2.8) ## End of Chapter Reflection You should know that... * Indices can be negative. * Irrational numbers are numbers that cannot be written as fractions. * The square root of a number that is not a perfect square is irrational. It is called a surd. * You can estimate the value of a surd by finding the two square (or cube) numbers closest to the number under the square (or cube) root. You should be able to... * Find the value of a number with a negative index. * Use the index laws for multiplication and division with positive, zero, and negative indices. * Recognise irrational numbers, including surds. * Estimate the value of a surd, stating which whole number it is closest to. Such as... * Write the value of 3^-2 * Write as a single power: a) 6³ x 6⁷ b) 11⁴ + 11^-4 * Sort the numbers into rational and irrational. -5.28, √18, ¹/₅, √-64, π, √118, 0.23 ³√38 * a) Estimate the value of √45 * b) Estimate the value of √69