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Algebra 1: Topic 1 Notes Table of Contents 1. 2. 3. 4. 5. 6. 7. Order of Operations & Evaluating Expressions The Real Number System Properties of Real Numbers Simplifying Radicals Basic Exponent Properties One & Two-Step Equations Basic Multi-Step Equations Order of Operations & Evaluating Expressions Review: Order of Operations Please- Parentheses Excuse- Exponents My- Multiplication Dear- Division Aunt- Addition Sally- Subtraction Review: Order of Operations Remember that Multiplication and Division are a pair…do in order from left to right. Addition and Subtraction are also a pair….do in order from left to right. Review: “Evaluate” • Evaluating an expression means to plug in! • For example: 2n + 4 for n = 3 2(3) + 4 10 Let’s Practice… Evaluate the following expressions. 1. 3r – 17 for r = 6 2. 10 – 6d for d = 4 3. 5g + 20 – (4g) for g = 5 4. (3 + p)2 – 2p for p = 7 The Real Number System: Numbers Have Names?!?! Natural Numbers • Non-decimal, positive numbers starting with one. Whole Numbers • Non-decimal, positive numbers and zero. Integers • Non-decimal positive and negative numbers, including zero. Rational Numbers • Any number that can be expressed as the 𝑝 quotient or fraction of two integers. 𝑞 • YES: – Any integers – Any decimals that ends or repeats – Any fraction • NO: – Never ending decimals Irrational Numbers • Any number that can not be expressed as a fraction. • Usually a never-ending, non-repeating decimal. • Examples: 𝜋 2, 5 1.2658945625692…. Let’s Practice… Rational or Irrational. 1. 2 17 2. 3. 4. 22 3 27 1 3 Properties of Real Numbers List of Properties of Real Numbers • • • • • Commutative Associative Distributive Identity Inverse Commutative Properties Definition: Changing the order of the numbers in addition or multiplication will not change the result. Commutative Properties Property Addition Multiplication Numbers Algebra Associative Properties Definition: Changing the grouping of the numbers in addition or multiplication will not change the result. Associative Properties Property Addition Multiplication Numbers Algebra Distributive Property Multiplication distributes over addition. ab c ab ac 32 5 3 2 3 5 Additive Identity Property Definition: Zero preserves identities under addition. In other words, adding zero to a number does not change its value. Example: Multiplicative Identity Property Definition: The number 1 preserves identities under multiplication. In other words, multiplying a number by 1 does not change the value of the number. Example: Additive Inverse Property Definition: For each real number a there exists a unique real number –a such that their sum is zero. In other words, opposites add to zero. Example: Multiplicative Inverse Property Definition: For each real number a there 1 exists a unique real number 1 𝑎 such that their product is 1. a Example: Simplifying Radicals Radical Vocab How to Simplify Radicals 1. Make a factor tree of the radicand. 2. Circle all final factor pairs. 3. All circled pairs move outside the radical and become single value. 4. Multiply all values outside radical. 5. Multiply all final factors that were not circled. Place product under radical sign. Let’s Practice… 1. 225 2. 300 Let’s Practice… 3. 1 49 4. 120 How to Simplify Cubed Radicals 1. Make a factor tree of the radicand. 2. Circle all final factor groups of three. 3. All circled groups of three move outside the radical and become single value. 4. Multiply all values outside radical. 5. Multiply all final factors that were not circled. Place product under radical sign. Let’s Practice… 3 1. 375 3 2. 64 Let’s Practice… 3 3. 81 3 4. 256 Exponents Definition: Exponent • The exponent of a number says how many times to use that number in a multiplication. It is written as a small number to the right and above the base number. The Zero Exponent Rule • Any number (excluding zero) to the zero power is always equal to one. • Examples: 1000=1 1470=1 550 =1 Negative Power Rule Let’s Practice… 1. 5-2 2. 3 −2 4 3. (-3)-3 The One Exponent Rule • Any number (excluding zero) to the first power is always equal to that number. • Examples: a1 = a 71 = 7 531 = 53 Simplifying Exponential Expressions Step 1: Simplify (get rid of negatives and fractions, where possible). Step 2: Plug in. Let’s Practice… 8b3 4a−2 1. 4m3j-2 2. for m = -2 and j = 3 for b = 2 and a = 3 Solving 2-step Equations Inverse Operations Pairs Addition Subtraction Multiplication Division Exponents Radicals How to Solve 2-Step Equations 1. Use the opposite operation to move the letters to the left and numbers to the right. 2. Divide to get the variable alone. Example 5x - 18 = 22 Let’s Practice… 1. 10x – 7 = 77 2. 15 – 3m = 45 Let’s Practice… 3. 4y = 3y + 12 4. 9b = 17 – 8b Fractions… To get rid of fractions, multiply all other values by the reciprocal! Example: k 6 2 4 Let’s Practice… m 14 24 1. 10 2. 3 − 2 x 5 =6 Solving Multi-Step Equations How to solve Multi-Step Equations 1. Distribute. Bonus Step: Multiply by reciprocal/LCM, if fractions. 2. Combine Like Terms (ONLY on same side of equal sign). 3. Use the inverse operation to move numbers to the right. 4. Use the inverse operation to move variables to the left. 5. Divide. Example 3(4x – 10) = 26 Example 6(4 + 3x) - 10x = 30 - 2x Example 1 x – 8 = 4 + 2x 5 Let’s Practice… 4(5 + 2x) + 7x = -5 Let’s Practice… 3 𝑥 2 +6=− 7 2 + 5𝑥