Order of Operations and Evaluating Expressions Section 1-2 Part 1

Goals Goal Rubric To simplify expressions involving exponents. Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary Power Exponent Base Simplify

Definition A power expression has two parts, a base and an exponent. Exponent Base 103 Power expression

Power In the power expression 103, 10 is called the base and 3 is called the exponent or power. 10 3 The exponent, 3, tells how many times the base, 10, is used as a factor. The base, 10, is the number that is used as a factor. 103 means 10 • 10 • 10 103 = 1000

Definition Base – In a power expression, the base is the number that is multiplied repeatedly. Example: In x3, x is the base. The exponent says to multiply the base by itself 3 times; x3 = x ⋅ x ⋅ x.

Definition Exponent – In a power expression, the exponent tells the number of times the base is used as a factor. Example: 24 equals 2 ⋅ 2 ⋅ 2 ⋅ 2. If a number has an exponent of 2, the number is often called squared. For example, 42 is read “4 squared.” Similarly, a number with an exponent of is called “cubed.”

Powers When a number is raised to the second power, we usually say it is “squared.” The area of a square is s  s = s2, where s is the side length. s When a number is raised to the third power, we usually say it is “cubed.” The volume of a cube is s  s  s = s3, where s is the side length. s

Powers Words Multiplication Power Value There are no easy geometric models for numbers raised to exponents greater than 3, but you can still write them using repeated multiplication or with a base and exponent. Reading Exponents Words Multiplication Power Value 3 to the first power 3 31 3 3 to the second power, or 3 squared 3  3 32 9 3 to the third power, or 3 cubed 3  3  3 33 27 3 to the fourth power 3  3  3  3 34 81 3 to the fifth power 3  3  3  3  3 35 243

Caution! In the expression –5², 5 is the base because the negative sign is not in parentheses. In the expression (–2)³, –2 is the base because of the parentheses.

Definition Simplify – a numerical expression is simplified when it is replaced with its single numerical value. Example: The simplest form of 2 • 8 is 16. To simplify a power, you replace it with its simplest name. The simplest form of 23 is 8.

Example: Evaluating Powers Simplify each expression. A. (–6)3 (–6)(–6)(–6) Use –6 as a factor 3 times. –216 B. –102 Think of a negative sign in front of a power as multiplying by a –1. –1 • 10 • 10 Find the product of –1 and two 10’s. –100

Example: Evaluating Powers Simplify the expression. C. 2 9  Use as a factor 2 times. 2 9 = 4 81 2 9 

Your Turn: Evaluate each expression. a. (–5)3 (–5)(–5)(–5) Use –5 as a factor 3 times. –125 b. –62 Think of a negative sign in front of a power as multiplying by –1. –1  6  6 Find the product of –1 and two 6’s. –36

Your Turn: Evaluate the expression. c. 3 Use as a factor 3 times. 4 27 64

Example: Writing Powers Write each number as a power of the given base. A. 64; base 8 8  8 The product of two 8’s is 64. 82 B. 81; base –3 (–3)(–3)(–3)(–3) The product of four –3’s is 81. (–3)4

Your Turn: Write each number as a power of a given base. a. 64; base 4 4  4  4 The product of three 4’s is 64. 43 b. –27; base –3 (–3)(–3)(–3) The product of three (–3)’s is –27. –33

Order of Operations Rules for arithmetic and algebra expressions that describe what sequence to follow to evaluate an expression involving more than one operation.

Order of Operations Evaluate 7 + 4 • 3. Is your answer 33 or 19? You can get 2 different answers depending on which operation you did first. We want everyone to get the same answer so we must follow the order of operations.

Remember the phrase “Please Excuse My Dear Aunt Sally” or PEMDAS. ORDER OF OPERATIONS 1. Parentheses - ( ) or [ ] 2. Exponents or Powers 3. Multiply and Divide (from left to right) 4. Add and Subtract (from left to right)

The Rules Step 1: First perform operations that are within grouping symbols such as parenthesis (), brackets [], and braces {}, and as indicated by fraction bars. Parenthesis within parenthesis are called nested parenthesis (( )). If an expression contains more than one set of grouping symbols, evaluate the expression from the innermost set first. Step 2: Evaluate Powers (exponents) or roots. Step 3: Perform multiplication or division operations in order by reading the problem from left to right. Step 4: Perform addition or subtraction operations in order by reading the problem from left to right.

Order of Operations Method 2 Method 1 Performing operations using order of operations Performing operations left to right only Can you imagine what it would be like if calculations were performed differently by various financial institutions or what if doctors prescribed different doses of medicine using the same formulas and achieving different results? The rules for order of operations exist so that everyone can perform the same consistent operations and achieve the same results. Method 2 is the correct method.

Order of Operations: Example 1 Evaluate without grouping symbols Follow the left to right rule: First solve any multiplication or division parts left to right. Then solve any addition or subtraction parts left to right. Divide A good habit to develop while learning order of operations is to underline the parts of the expression that you want to solve first. Then rewrite the expression in order from left to right and solve the underlined part(s). Multiply Add The order of operations must be followed each time you rewrite the expression.

Order of Operations: Example 2 Expressions with powers Follow the left to right rule: First solve exponent/(powers). Second solve multiplication or division parts left to right. Then solve any addition or subtraction parts left to right. Exponents (powers) A good habit to develop while learning order of operations is to underline the parts of the expression that you want to solve first. Then rewrite the expression in order from left to right and solve the underlined part(s). Multiply Subtract The order of operations must be followed each time you rewrite the expression.

Order of Operations: Example 3 Evaluate with grouping symbols Follow the left to right rule: First solve parts inside grouping symbols according to the order of operations. Solve any exponent/(Powers). Then solve multiplication or division parts left to right. Then solve any addition or subtraction parts left to right. Grouping symbols Subtract A good habit to develop while learning order of operations is to underline the parts of the expression that you want to solve first. Then rewrite the expression in order from left to right and solve the underlined part(s). Exponents (powers) Multiply Divide The order of operations must be followed each time you rewrite the expression.

Order of Operations: Example 4 Expressions with fraction bars A good habit to develop while learning order of operations is to underline the parts of the expression that you want to solve first. Then rewrite the expression in order from left to right and solve the underlined part(s). Exponents (powers) Work above the fraction bar Multiply Work below the fraction bar Grouping symbols Subtract Follow the left to right rule: Follow the order of operations by working to solve the problem above the fraction bar. Then follow the order of operations by working to solve the problem below the fraction bar. Finally, recall that fractions are also division problems – simplify the fraction. Simplify: Divide Add The order of operations must be followed each time you rewrite the expression.

Your Turn: Simplify the expression. 8 ÷ · 3 8 ÷ · 3 8 ÷ · 3 1 2 8 ÷ · 3 1 2 There are no grouping symbols. 16 · 3 Divide. 48 Multiply.

Your Turn: Simplify the expression. 5.4 – 32 + 6.2 There are no grouping symbols. 5.4 – 32 + 6.2 5.4 – 9 + 6.2 Simplify powers. –3.6 + 6.2 Subtract 2.6 Add.

Your Turn: Simplify the expression. –20 ÷ [–2(4 + 1)] There are two sets of grouping symbols. –20 ÷ [–2(4 + 1)] Perform the operations in the innermost set. –20 ÷ [–2(5)] Perform the operation inside the brackets. –20 ÷ –10 2 Divide.

Your Turn: Which of the following represents 112 + 18 - 33 · 5 in simplified form? -3,236 4 107 16,996

Your Turn: Simplify 16 - 2(10 - 3) 2 -7 12 98

Your Turn: Simplify 24 – 6 · 4 ÷ 2 72 36 12

Caution! Fraction bars, radical symbols, and absolute-value symbols can also be used as grouping symbols. Remember that a fraction bar indicates division.

Your Turn: Simplify. 5 + 2(–8) (–2) – 3 (–2) – 3 3 The fraction bar acts as a grouping symbol. Simplify the numerator and the denominator before dividing. 5 + 2(–8) (–2) – 3 3 5 + 2(–8) –8 – 3 Evaluate the power in the denominator. 5 + (–16) – 8 – 3 Multiply to simplify the numerator. –11 Add. 1 Divide.

Your Turn: Simplify. 2(–4) + 22 42 – 9 The fraction bar acts as a grouping symbol. Simplify the numerator and the denominator before dividing. 2(–4) + 22 42 – 9 –8 + 22 42 – 9 Multiply to simplify the numerator. –8 + 22 16 – 9 Evaluate the power in the denominator. Add to simplify the numerator. Subtract to simplify the denominator. 14 7 2 Divide.

Joke Time Why did the chicken get into trouble? Because she used fowl language. Why don’t anteaters get sick? Because they are full of ant-ibodies. What do you call a really old ant? An Antique.

Assignment 1.2 Part 1 Exercises Pg. 14: #9 – 33