1.The set of integers

# 1.The set of integers ## Diagnostic assessment ### Introduction This lesson is designed to help us understand what you already know about integers. We will review the definition of integers, how to represent them, compare them, and perform basic operations like addition, subtraction, multiplication, and division. This will help us identify areas where you might need more practice. ### Key Concepts #### What are Integers? Integers are whole numbers and their opposites. They include: * **Positive Integers:** \( 1, 2, 3, ... \) (Natural numbers) * **Zero:** \( 0 \) * **Negative Integers:** \( -1, -2, -3, ... \) Integers do not include fractions or decimals. #### Representing Integers ##### The Number Line The number line is a visual tool used to represent integers. * Zero is at the center. * Positive integers are to the right of zero. * Negative integers are to the left of zero. \[ \text{... -3 -2 -1 0 1 2 3 ...} \] #### Comparing and Ordering Integers * Numbers to the right on the number line are greater. * Numbers to the left on the number line are smaller. * We use symbols: * \( > \) (greater than) * \( < \) (less than) * \( = \) (equal to) For example, \( 3 > -5 \) and \( -2 < 1 \). #### Operations with Integers ##### Addition of Integers * **Same Signs:** Add the absolute values and keep the common sign. * Example: \( 5 + 3 = 8 \) * Example: \( -5 + (-3) = -8 \) * **Different Signs:** Subtract the smaller absolute value from the larger absolute value. The sign of the result is the same as the number with the larger absolute value. * Example: \( 5 + (-3) = 2 \) (since \( |5| > |-3| \) and 5 is positive) * Example: \( -5 + 3 = -2 \) (since \( |-5| > |3| \) and -5 is negative) ##### Subtraction of Integers To subtract an integer, add its additive inverse (or opposite). \( a - b = a + (-b) \) * Example: \( 7 - 4 = 7 + (-4) = 3 \) * Example: \( 7 - (-4) = 7 + 4 = 11 \) * Example: \( -7 - 4 = -7 + (-4) = -11 \) ##### Multiplication of Integers * **Same Signs:** The product is positive. * Example: \( 3 \times 4 = 12 \) * Example: \( (-3) \times (-4) = 12 \) * **Different Signs:** The product is negative. * Example: \( 3 \times (-4) = -12 \) * Example: \( (-3) \times 4 = -12 \) ##### Division of Integers * **Same Signs:** The quotient is positive. * Example: \( 12 \div 4 = 3 \) * Example: \( (-12) \div (-4) = 3 \) * **Different Signs:** The quotient is negative. * Example: \( 12 \div (-4) = -3 \) * Example: \( (-12) \div 4 = -3 \) * **Division by zero is undefined.** #### Absolute Value The absolute value of an integer is its distance from zero on the number line, always a non-negative value. It is denoted by vertical bars \( |\text{number}| \). * Example: \( |5| = 5 \) * Example: \( |-5| = 5 \) ### Vocabulary * **Integer:** A whole number (not a fraction or decimal) that can be positive, negative, or zero. * **Positive Integer:** An integer greater than zero (e.g., \( 1, 2, 3 \)). * **Negative Integer:** An integer less than zero (e.g., \( -1, -2, -3 \)). * **Zero:** The integer that is neither positive nor negative. * **Number Line:** A line on which numbers are marked at regular intervals, used to illustrate numerical operations and relationships. * **Absolute Value:** The distance of a number from zero on the number line, always expressed as a positive value. * **Additive Inverse (Opposite):** Two numbers are additive inverses if their sum is zero (e.g., the additive inverse of 5 is -5). ### Examples 1. **Ordering:** Arrange the following integers in ascending order: \( -8, 2, -1, 0, 5, -3 \) * Solution: On a number line, \( -8 \) is furthest left, then \( -3, -1, 0, 2, 5 \). * So, ascending order is: \( -8, -3, -1, 0, 2, 5 \). 2. **Addition:** Calculate \( -12 + 7 \) * Solution: The signs are different. Subtract \( |7| \) from \( |-12| \): \( 12 - 7 = 5 \). Since \( |-12| > |7| \) and -12 is negative, the answer is negative. * Answer: \( -5 \). 3. **Subtraction:** Calculate \( 9 - (-3) \) * Solution: Change subtraction to addition of the opposite: \( 9 + 3 \). * Answer: \( 12 \). 4. **Multiplication:** Calculate \( (-6) \times 5 \) * Solution: Signs are different, so the product is negative. \( 6 \times 5 = 30 \). * Answer: \( -30 \). 5. **Division:** Calculate \( -24 \div (-8) \) * Solution: Signs are the same, so the quotient is positive. \( 24 \div 8 = 3 \). * Answer: \( 3 \). 6. **Absolute Value:** Find \( |-15| \) and \( |10| \). * Solution: The distance of -15 from zero is 15. The distance of 10 from zero is 10. * Answer: \( |-15| = 15 \) and \( |10| = 10 \). ### Exercises 1. Place \( <, >, \) or \( = \) in the blank: a) \( -5 \text{ \_\_\_ } 2 \) b) \( -10 \text{ \_\_\_ } -12 \) c) \( | -7 | \text{ \_\_\_ } 7 \) 2. Calculate the following: a) \( -8 + (-4) \) b) \( 15 + (-6) \) c) \( 10 - (-3) \) d) \( -18 - 5 \) e) \( 7 \times (-9) \) f) \( (-11) \times (-3) \) g) \( 36 \div (-9) \) h) \( (-40) \div (-5) \) 3. Evaluate: a) \( | -23 | \) b) \( | 9 | \) ### Assignment 1. Order the following integers from least to greatest: \( 1, -6, 0, 4, -10, -2 \) 2. Perform the indicated operations: a) \( -14 + 9 \) b) \( -20 + (-10) \) c) \( 18 - 25 \) d) \( -5 - (-13) \) e) \( 8 \times (-7) \) f) \( (-9) \times (-6) \) g) \( 54 \div (-6) \) h) \( (-72) \div 8 \) 3. A submarine is at a depth of \( -250 \) meters. It ascends \( 100 \) meters. What is its new depth? Write an expression and solve. 4. The temperature dropped from \( 5^\circ\text{C} \) to \( -3^\circ\text{C} \). What was the change in temperature? ### Summary Today, we reviewed the fundamental concepts of integers. We learned that integers include positive numbers, negative numbers, and zero. We practiced representing them on a number line, comparing their values, and performing all four basic arithmetic operations (addition, subtraction, multiplication, and division) while paying close attention to the rules for signs. We also revisited the concept of absolute value, which is the distance of a number from zero. Understanding these basic rules is crucial for more advanced mathematical topics. ## Operations in the set of integers ### Introduction Welcome to this lesson on operations within the set of integers! You've learned about natural numbers and whole numbers. Now, we'll explore integers, which include positive numbers, negative numbers, and zero. Understanding how to add, subtract, multiply, and divide integers is fundamental to higher-level mathematics. ### Key Concepts #### The Set of Integers Integers are all the whole numbers and their opposites. They can be represented as: \[ \{..., -3, -2, -1, 0, 1, 2, 3, ...\} \] * **Positive Integers:** Numbers greater than zero (e.g., \( 1, 2, 3 \)). * **Negative Integers:** Numbers less than zero (e.g., \( -1, -2, -3 \)). * **Zero:** Neither positive nor negative. Integers can be visualized on a number line: ``` <---|-5-|-4-|-3-|-2-|-1-|0|1|2|3|4|5|---> ``` #### Addition of Integers ##### 1. Adding Integers with the Same Sign * **Rule:** Add their absolute values and keep the common sign. * **Examples:** * \( 3 + 5 = 8 \) * \( (-3) + (-5) = -8 \) ##### 2. Adding Integers with Different Signs * **Rule:** Subtract the smaller absolute value from the larger absolute value, and use the sign of the number with the larger absolute value. * **Examples:** * \( -7 + 3 = -4 \) (\( | -7 | = 7 \), \( | 3 | = 3 \); \( 7-3=4 \); sign of -7 is negative) * \( 7 + (-3) = 4 \) (\( | 7 | = 7 \), \( | -3 | = 3 \); \( 7-3=4 \); sign of 7 is positive) ##### 3. Adding Zero * **Rule:** Adding zero to any integer does not change its value. * **Example:** \( a + 0 = a \) #### Subtraction of Integers * **Rule:** Subtracting an integer is the same as adding its opposite (additive inverse). \[ a - b = a + (-b) \] * **Examples:** * \( 5 - 3 = 5 + (-3) = 2 \) * \( 3 - 5 = 3 + (-5) = -2 \) * \( 5 - (-3) = 5 + 3 = 8 \) (The opposite of -3 is 3) * \( -4 - 6 = -4 + (-6) = -10 \) #### Multiplication of Integers ##### 1. Multiplying Integers with the Same Sign * **Rule:** The product is always positive. * Positive \( \times \) Positive \( = \) Positive (e.g., \( 3 \times 4 = 12 \)) * Negative \( \times \) Negative \( = \) Positive (e.g., \( (-3) \times (-4) = 12 \)) ##### 2. Multiplying Integers with Different Signs * **Rule:** The product is always negative. * Positive \( \times \) Negative \( = \) Negative (e.g., \( 3 \times (-4) = -12 \)) * Negative \( \times \) Positive \( = \) Negative (e.g., \( (-3) \times 4 = -12 \)) ##### 3. Multiplying by Zero * **Rule:** Any integer multiplied by zero is zero. * **Example:** \( a \times 0 = 0 \) #### Division of Integers ##### 1. Dividing Integers with the Same Sign * **Rule:** The quotient is always positive. * Positive \( \div \) Positive \( = \) Positive (e.g., \( 12 \div 3 = 4 \)) * Negative \( \div \) Negative \( = \) Positive (e.g., \( (-12) \div (-3) = 4 \)) ##### 2. Dividing Integers with Different Signs * **Rule:** The quotient is always negative. * Positive \( \div \) Negative \( = \) Negative (e.g., \( 12 \div (-3) = -4 \)) * Negative \( \div \) Positive \( = \) Negative (e.g., \( (-12) \div 3 = -4 \)) ##### 3. Division by Zero * **Rule:** Division by zero is undefined. You cannot divide any number by zero. * **Example:** \( a \div 0 \) is undefined. #### Order of Operations When you have an expression with multiple operations, you must follow a specific order. This is often remembered using the acronym BODMAS or PEMDAS: * **B**rackets (or **P**arentheses) * **O**rders (or **E**xponents/Powers) * **D**ivision and **M**ultiplication (from left to right) * **A**ddition and **S**ubtraction (from left to right) ### Vocabulary * **Integers:** The set of whole numbers and their opposites, including zero. \( \{..., -3, -2, -1, 0, 1, 2, 3, ...\} \) * **Positive Integer:** An integer greater than zero. * **Negative Integer:** An integer less than zero. * **Absolute Value:** The distance of a number from zero on the number line, always positive. Denoted by \( |a| \). For example, \( | -5 | = 5 \) and \( | 5 | = 5 \). * **Opposite (Additive Inverse):** Two numbers are opposites if their sum is zero (e.g., 5 and -5). * **Sum:** The result of an addition. * **Difference:** The result of a subtraction. * **Product:** The result of a multiplication. * **Quotient:** The result of a division. * **Order of Operations:** The set of rules that dictate the sequence in which operations should be performed in a mathematical expression. ### Examples 1. **Addition:** * Calculate \( -8 + 5 \) * *Solution:* Signs are different. Subtract absolute values: \( 8 - 5 = 3 \). The number with the larger absolute value is -8, so the sign is negative. \( -8 + 5 = -3 \) * Calculate \( -12 + (-7) \) * *Solution:* Signs are the same. Add absolute values: \( 12 + 7 = 19 \). Keep the common sign, which is negative. \( -12 + (-7) = -19 \) 2. **Subtraction:** * Calculate \( 6 - (-9) \) * *Solution:* Change subtraction to addition of the opposite: \( 6 + 9 = 15 \) * Calculate \( -10 - 4 \) * *Solution:* Change subtraction to addition of the opposite: \( -10 + (-4) = -14 \) 3. **Multiplication:** * Calculate \( -5 \times 6 \) * *Solution:* Different signs, so the product is negative: \( 5 \times 6 = 30 \), so \( -5 \times 6 = -30 \) * Calculate \( -7 \times (-8) \) * *Solution:* Same signs, so the product is positive: \( 7 \times 8 = 56 \), so \( -7 \times (-8) = 56 \) 4. **Division:** * Calculate \( 24 \div (-6) \) * *Solution:* Different signs, so the quotient is negative: \( 24 \div 6 = 4 \), so \( 24 \div (-6) = -4 \) * Calculate \( -45 \div (-9) \) * *Solution:* Same signs, so the quotient is positive: \( 45 \div 9 = 5 \), so \( -45 \div (-9) = 5 \) 5. **Mixed Operations (Order of Operations):** * Calculate \( (3 - 7) \times (-2) + 10 \div (-5) \) * *Solution:* 1. **Brackets:** \( (3 - 7) = 3 + (-7) = -4 \) 2. Now the expression is: \( -4 \times (-2) + 10 \div (-5) \) 3. **Multiplication/Division (from left to right):** * \( -4 \times (-2) = 8 \) * \( 10 \div (-5) = -2 \) 4. Now the expression is: \( 8 + (-2) \) 5. **Addition:** \( 8 + (-2) = 6 \) * *Final Answer:* \( 6 \) ### Exercises Solve the following problems. Show your working. 1. \( -15 + 8 \) 2. \( 12 + (-20) \) 3. \( (-6) + (-11) \) 4. \( 7 - 13 \) 5. \( -9 - (-4) \) 6. \( 2 - (-10) \) 7. \( 8 \times (-3) \) 8. \( (-5) \times (-9) \) 9. \( -24 \div 6 \) 10. \( 36 \div (-4) \) 11. \( (-40) \div (-8) \) 12. \( (10 - 15) \times 3 + (-20) \div 4 \) 13. \( 18 \div (-2) + (-3) \times (-5) \) ### Assignment Practice these problems at home. 1. What is the sum of -25 and 15? 2. Subtract -7 from -12. 3. Find the product of 11 and -4. 4. Divide -63 by -9. 5. Evaluate \( -5 + (-3) - 10 \). 6. Evaluate \( 7 \times (4 - 9) + 25 \div 5 \). 7. A submarine is at 200 meters below sea level (represented as -200 m). It ascends 75 meters. What is its new depth? 8. The temperature in a city dropped from \( 5^ ext{o}C \) to \( -8^ ext{o}C \) overnight. What was the change in temperature? ### Summary In this lesson, we have learned the essential rules for performing operations with integers: * **Addition:** * Same signs: Add absolute values, keep the sign. * Different signs: Subtract absolute values, use the sign of the number with the larger absolute value. * **Subtraction:** Change subtraction to addition of the opposite.\( a - b = a + (-b) \) * **Multiplication and Division:** * Same signs (\( +,+ \) or \( -,- \)): Result is positive. * Different signs (\( +,- \) or \( -,+ \)): Result is negative. * **Division by zero is undefined.** * Always remember to follow the **Order of Operations (BODMAS/PEMDAS)** when solving expressions with multiple operations. ## Whole number powers. ### Introduction Welcome to our lesson on **whole number powers**! In mathematics, we often need to multiply the same number by itself many times. Writing this out repeatedly can be long and tedious. Powers provide a neat and short way to express these repeated multiplications. Think of it as a shorthand that makes calculations and expressions much simpler. ### Key Concepts #### Understanding Powers A power is a way to show that a number is multiplied by itself a certain number of times. It has two main parts: * **The Base:** This is the number that is being multiplied by itself. * **The Exponent (or Index):** This is a small number written above and to the right of the base. It tells us how many times the base is used as a factor in the multiplication. #### Notation When we write a number in power form, it looks like this: \[ a^n \] * Here, \( a \) is the **base**. * \( n \) is the **exponent**. This means we multiply the base \( a \) by itself \( n \) times. So, \( a^n = a \times a \times a \times \ldots \times a \) (where \( a \) appears \( n \) times). #### Reading Powers We have special ways to read powers, especially for exponents 2 and 3: * \( a^2 \) is read as "a squared" or "a to the power of 2". * \( a^3 \) is read as "a cubed" or "a to the power of 3". * For any other exponent, \( a^n \) is read as "a to the power of n" or "a to the nth power". #### Evaluating Powers To evaluate (or calculate the value of) a power, you multiply the base by itself the number of times indicated by the exponent. #### Powers of 0 and 1 There are two important rules for powers involving 0 and 1: * **Any number to the power of 1:** Any number (except 0) raised to the power of 1 is the number itself. \[ a^1 = a \] * **Any non-zero number to the power of 0:** Any non-zero number raised to the power of 0 is always 1. \[ a^0 = 1 \quad (\text{where } a \neq 0) \] #### Powers of Negative Numbers When the base is a negative number, the sign of the result depends on the exponent: * If the exponent is **even**, the result will be **positive**. Example: \( (-2)^4 = (-2) \times (-2) \times (-2) \times (-2) = 16 \) * If the exponent is **odd**, the result will be **negative**. Example: \( (-2)^3 = (-2) \times (-2) \times (-2) = -8 \) **Important Note:** Pay close attention to parentheses! \( -2^4 \) means \( -(2 \times 2 \times 2 \times 2) = -16 \), while \( (-2)^4 \) means \( (-2) \times (-2) \times (-2) \times (-2) = 16 \). #### Order of Operations (BODMAS/PEMDAS) with Powers When you have an expression with different operations, always remember the order: 1. **B**rackets (or **P**arentheses) 2. **O**rders (or **E**xponents/Powers) 3. **D**ivision and **M**ultiplication (from left to right) 4. **A**ddition and **S**ubtraction (from left to right) Powers are evaluated **before** multiplication, division, addition, or subtraction, unless brackets indicate otherwise. ### Vocabulary * **Power:** A mathematical expression indicating repeated multiplication of a number by itself (e.g., \( 3^4 \)). * **Base:** The number that is multiplied by itself in a power (e.g., in \( 3^4 \), 3 is the base). * **Exponent (or Index):** The small number written above and to the right of the base, indicating how many times the base is used as a factor (e.g., in \( 3^4 \), 4 is the exponent). * **Squared:** A number raised to the power of 2 (e.g., \( x^2 \)). * **Cubed:** A number raised to the power of 3 (e.g., \( x^3 \)). ### Examples Let's work through some examples to understand how to evaluate powers: 1. **Evaluate \( 5^2 \).** \( 5^2 = 5 \times 5 = 25 \) 2. **Evaluate \( 3^4 \).** \( 3^4 = 3 \times 3 \times 3 \times 3 = 81 \) 3. **Evaluate \( 10^0 \).** According to the rule, any non-zero number to the power of 0 is 1. \( 10^0 = 1 \) 4. **Evaluate \( (-3)^3 \).** The base is -3 and the exponent is odd, so the result will be negative. \( (-3)^3 = (-3) \times (-3) \times (-3) = 9 \times (-3) = -27 \) 5. **Evaluate \( -3^2 \).** Here, only 3 is being squared, not -3. The negative sign is applied after the power is calculated. \( -3^2 = -(3 \times 3) = -9 \) 6. **Evaluate \( 2 \times 3^2 \).** Follow BODMAS: powers first, then multiplication. \( 2 \times 3^2 = 2 \times (3 \times 3) = 2 \times 9 = 18 \) 7. **Evaluate \( (4-2)^3 \).** Follow BODMAS: brackets first, then powers. \( (4-2)^3 = (2)^3 = 2 \times 2 \times 2 = 8 \) ### Exercises Test your understanding with these exercises: 1. Evaluate the following powers: a. \( 7^2 \) b. \( 2^5 \) c. \( 1^8 \) d. \( 9^0 \) e. \( (-5)^2 \) f. \( -5^2 \) 2. Simplify the following expressions: a. \( 3^2 + 4^2 \) b. \( 2 \times 5^3 - 10 \) ### Assignment Practice these questions at home to strengthen your skills: 1. Evaluate: a. \( 6^3 \) b. \( 11^2 \) c. \( (-4)^3 \) d. \( 8^1 \) e. \( 12^0 \) 2. Calculate the value of the following: a. \( (10-7)^4 \) b. \( 5 \times 2^4 + 3 \times 1^5 \) 3. A square garden has sides of length 7 meters. What is its area using powers? ### Summary In this lesson, we've learned about whole number powers, a powerful way to represent repeated multiplication. Remember these key points: * A power consists of a **base** and an **exponent** (\( a^n \)). * The exponent tells you how many times to multiply the base by itself. * \( a^2 \) is called "a squared," and \( a^3 \) is called "a cubed." * Any number to the power of 1 is the number itself (\( a^1 = a \)). * Any non-zero number to the power of 0 is 1 (\( a^0 = 1 \)). * Be careful with negative bases: even exponents result in positive answers, while odd exponents result in negative answers (e.g., \( (-2)^4 = 16 \) but \( (-2)^3 = -8 \)). Also, distinguish between \( -a^n \) and \( (-a)^n \). * Always follow the order of operations (BODMAS/PEMDAS) when evaluating expressions involving powers. Understanding powers is fundamental for more advanced topics in algebra and beyond. Keep practicing! ## Laws of indices # Laws of indices\n\n## Introduction\n\nWelcome to the exciting world of indices! In mathematics, we often encounter situations where a number is multiplied by itself many times. For example, \( 2 \times 2 \times 2 \times 2 \times 2 \). Writing this repeatedly can be long and tedious. Indices provide a powerful and shorthand way to represent such repeated multiplication, making calculations and expressions much simpler.\n\nIn this lesson, you will learn what indices are, identify their key parts (base and exponent), and most importantly, understand and apply the fundamental laws that govern operations with indices. These laws will help you simplify complex expressions and solve problems more efficiently.\n\n## Key Concepts\n\n### Understanding Indices\n\nIndices (also called exponents or powers) are a way to show that a number is multiplied by itself a certain number of times.\n\n* **Base:** The number that is being multiplied repeatedly.\n* **Exponent (or Index):** The small number written above and to the right of the base. It tells you how many times the base is multiplied by itself.\n* **Power:** The entire expression, consisting of the base and the exponent.\n\nFor example, in the expression \( 5^3 \):\n* The **base** is 5.\n* The **exponent** (or index) is 3.\n* The **power** is \( 5^3 \), read as "5 to the power of 3" or "5 cubed", which means \( 5 \times 5 \times 5 \).\n\n### The Laws of Indices\n\nThere are several important laws that allow us to manipulate and simplify expressions involving indices. Let's explore each one:\n\n#### 1. Product Law (Multiplication Law)\n\nWhen multiplying numbers with the same base, you add their exponents.\n\n* **Formula:** \[ a^m \times a^n = a^{m+n} \]\n\n#### 2. Quotient Law (Division Law)\n\nWhen dividing numbers with the same base, you subtract their exponents.\n\n* **Formula:** \[ a^m \div a^n = a^{m-n} \] (where \( a \neq 0 \))\n\n#### 3. Power Law (Power of a Power Law)\n\nWhen an expression with an exponent is raised to another exponent, you multiply the exponents.\n\n* **Formula:** \[ (a^m)^n = a^{mn} \]\n\n#### 4. Zero Index Law\n\nAny non-zero number raised to the power of zero is always equal to 1.\n\n* **Formula:** \[ a^0 = 1 \] (where \( a \neq 0 \))\n\n#### 5. Negative Index Law\n\nA number raised to a negative exponent is equal to the reciprocal of the number raised to the positive exponent.\n\n* **Formula:** \[ a^{-n} = \frac{1}{a^n} \] (where \( a \neq 0 \))\n\n#### 6. Power of a Product Law\n\nWhen a product of numbers is raised to a power, you apply the exponent to each factor in the product.\n\n* **Formula:** \[ (ab)^n = a^n b^n \]\n\n#### 7. Power of a Quotient Law\n\nWhen a fraction (quotient) is raised to a power, you apply the exponent to both the numerator and the denominator.\n\n* **Formula:** \[ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \] (where \( b \neq 0 \))\n\n## Vocabulary\n\n* **Index (plural: Indices):** The small number written above and to the right of the base, indicating how many times the base is multiplied by itself. Also known as **exponent** or **power** (when referring to the small number itself).\n* **Exponent:** See **Index**.\n* **Base:** The number that is being multiplied repeatedly in an index expression.\n* **Power:** The entire expression \( a^n \). For example, \( 2^3 \) is "2 to the power of 3".\n* **Reciprocal:** For a number \( x \) (where \( x \neq 0 \)), its reciprocal is \( \frac{1}{x} \). For example, the reciprocal of 4 is \( \frac{1}{4} \).\n\n## Examples\n\nLet's apply the laws of indices to simplify various expressions.\n\n1. **Product Law:**\n * Simplify \( 3^2 \times 3^4 \).\n \[ 3^2 \times 3^4 = 3^{2+4} = 3^6 \]\n * Simplify \( x^5 \times x^3 \).\n \[ x^5 \times x^3 = x^{5+3} = x^8 \]\n\n2. **Quotient Law:**\n * Simplify \( 7^8 \div 7^3 \).\n \[ 7^8 \div 7^3 = 7^{8-3} = 7^5 \]\n * Simplify \( m^9 \div m^2 \).\n \[ m^9 \div m^2 = m^{9-2} = m^7 \]\n\n3. **Power Law:**\n * Simplify \( (4^3)^2 \).\n \[ (4^3)^2 = 4^{3 \times 2} = 4^6 \]\n * Simplify \( (y^4)^5 \).\n \[ (y^4)^5 = y^{4 \times 5} = y^{20} \]\n\n4. **Zero Index Law:**\n * Simplify \( 10^0 \).\n \[ 10^0 = 1 \]\n * Simplify \( (5xy)^0 \).\n \[ (5xy)^0 = 1 \] (provided \( 5xy \neq 0 \))\n\n5. **Negative Index Law:**\n * Simplify \( 2^{-3} \).\n \[ 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \]\n * Simplify \( a^{-5} \).\n \[ a^{-5} = \frac{1}{a^5} \]\n * Express \( \frac{1}{x^4} \) using a negative index.\n \[ \frac{1}{x^4} = x^{-4} \]\n\n6. **Power of a Product Law:**\n * Simplify \( (3x)^2 \).\n \[ (3x)^2 = 3^2 x^2 = 9x^2 \]\n * Simplify \( (2ab)^3 \).\n \[ (2ab)^3 = 2^3 a^3 b^3 = 8a^3b^3 \]\n\n7. **Power of a Quotient Law:**\n * Simplify \( \left(\frac{2}{3}\right)^3 \).\n \[ \left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27} \]\n * Simplify \( \left(\frac{x}{y^2}\right)^4 \).\n \[ \left(\frac{x}{y^2}\right)^4 = \frac{x^4}{(y^2)^4} = \frac{x^4}{y^{2 \times 4}} = \frac{x^4}{y^8} \]\n\n8. **Combined Laws:**\n * Simplify \( \frac{(x^3 y^2)^2}{x^4 y} \).\n \[ \frac{(x^3 y^2)^2}{x^4 y} = \frac{x^{3 \times 2} y^{2 \times 2}}{x^4 y} = \frac{x^6 y^4}{x^4 y^1} = x^{6-4} y^{4-1} = x^2 y^3 \]\n\n## Exercises\n\nSimplify the following expressions using the laws of indices:\n\n1. \( 4^3 \times 4^5 \)\n2. \( p^7 \div p^3 \)\n3. \( (y^5)^2 \)\n4. \( 15^0 \)\n5. \( 5^{-2} \)\n6. \( (3m)^4 \)\n7. \( \left(\frac{a}{b}\right)^5 \)\n8. \( 2^3 \times 2^0 \times 2^{-1} \)\n9. \( \frac{(x^4)^2}{x^3} \)\n10. \( (2a^2b^3)^3 \)\n\n## Assignment\n\nSolve the following problems, showing all your steps clearly.\n\n1. Simplify \( \frac{6^7 \times 6^2}{6^4} \).\n2. Evaluate \( 3^{-2} + 2^{-1} \).\n3. Simplify \( \frac{(m^5 n^3)^2}{m^2 n^4} \).\n4. If \( x = 2 \) and \( y = 3 \), evaluate \( (x^2y^{-1})^2 \).\n5. Express \( \frac{1}{5^{-3}} \) in positive index form.\n6. Simplify \( \left(\frac{2x^2}{y^3}\right)^3 \times \frac{y^7}{4x^5} \).\n\n## Summary\n\nCongratulations! You've learned the fundamental laws of indices. Remember these key points:\n\n* Indices offer a shorthand for repeated multiplication: \( a^n \) means \( a \) multiplied by itself \( n \) times.\n* **Product Law:** Add exponents when multiplying powers with the same base (\( a^m \times a^n = a^{m+n} \)).\n* **Quotient Law:** Subtract exponents when dividing powers with the same base (\( a^m \div a^n = a^{m-n} \)).\n* **Power Law:** Multiply exponents when raising a power to another power (\( (a^m)^n = a^{mn} \)).\n* **Zero Index Law:** Any non-zero number to the power of zero is 1 (\( a^0 = 1 \)).\n* **Negative Index Law:** A negative exponent means the reciprocal of the base raised to the positive exponent (\( a^{-n} = \frac{1}{a^n} \)).\n* **Power of a Product/Quotient:** Apply the exponent to each term inside the bracket (e.g., \( (ab)^n = a^n b^n \) and \( (a/b)^n = a^n/b^n \)).\n\nMastering these laws will significantly improve your ability to handle algebraic expressions and solve a wide range of mathematical problems. Keep practicing!