We say, 2 to the fourth power is equal to 16. You Try It 1: Identify the base and exponent in each expression and simplify.

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1 Section.5 Exponents and Order of Operations.5 Exponents and Order of Operations Recall Exponents are We a say, shorthand 4 to the way third of expressing power is equal repeated to 64. multiplication. 4 6 Here, is called the base and 4 is called the exponent. We say, to the fourth power is equal to Here, 4 is called the base and is called the exponent. Example : Identify the base and exponent in each expression and simplify. a) 9 b) 6 Solution: a) In 9, 9 is the base and is the exponent b) In 6, 6 is the base and is the exponent You Try It : Identify the base and exponent in each expression and simplify. a) 5 b) 5 Example : Translate the following English phrases into numerical expressions. a) seven to the fifth power b) twelve to the first power 5 Solution: a) seven to the fifth power 7 b) twelve to the first power 05 Carreon

2 Section.5 Exponents and Order of Operations You Try It : Translate the following English phrases into numerical expressions. a) five to the eighth power b) ten to the third power Squared In expressions with the exponent is, the base is said to be squared. Example : Translate the following numerical expressions to English. a) 9 b) 9 c) 9 Solution: a) We say nine squared or the square of nine to represent 9. b) We say the square of negative nine to represent 9. c) We say the opposite of the square of nine to represent 9. CAUTION: Using negative nine squared to represent 9 is incorrect since it is not clear whether it represents 9 or 9. Later, we will learn that these two expressions give us different answers. We avoid using negative nine squared altogether because it is ambiguous. You Try It : Translate the following numerical expressions to English. a) 5 b) 5 c) 5 Cubed In expressions with the exponent is, the base is said to be cubed. 05 Carreon

3 Section.5 Exponents and Order of Operations Example 4: Translate the following numerical expressions to English. a) 4 b) 4 c) 4 Solution: a) We say four cubed or the cube of four to represent 4. b) We say the cube of negative four to represent 4. c) We say the opposite of the cube of four to represent 4. CAUTION: Using negative four cubed to represent 4 is incorrect since it is not clear whether it represents 4 or 4 4. Later, we will learn that and 4 will give us the same answers but we still avoid using negative four cubed altogether because it is ambiguous in its translation. You Try It 4: Translate the following numerical expressions to English. a) 6 b) 6 c) 6 What if we don t see an exponent? If there is no exponent written with a number, it is implied that the exponent is one. What if the exponent is zero? If a non-zero number is written with a zero exponent the result is always. CAUTION: Exponents only apply to the number or letter directly below/to the left of them. The presence of parentheses indicate if an exponent is to be applied to a sign or more than one factor at a time. 05 Carreon

4 Section.5 Exponents and Order of Operations Example 5: Simplify each expression. a) 0 5 b) 7 c) d) 8 0 Solution: 0 a) The exponent is zero so 5. b) The exponent is one so 7 7. c) The exponent is one so. d) The exponent is zero so 0 8 You Try It 5: Simplify each expression. a) b) 9 0 c) 4 0 d) 5 Powers of Negative Numbers Recall the following facts from section.4: The product of an even number of negative factors is positive. The product of an odd number of negative factors is negative. Since exponents mean repeated multiplication of the same base, we extend our findings above to figure out the sign of our answers when raising negative numbers to even or odd exponents. A negative number, raised to an even exponent of gives us a positive result. A negative number, raised to an odd exponent of gives us a negative result. 05 Carreon 4

5 Section.5 Exponents and Order of Operations Example 6: Decide whether each of the following gives a positive or negative result. Then simplify. a) b) c) 7 d) 0 Solution: a) The exponent,, is even so the answer is positive.. b) The exponent,, is odd so the answer is negative.. c) The exponent, 7, is odd so the answer is negative d) The exponent, 0, is even so the answer is positive You Try It 6: Decide whether each of the following gives a positive or negative result. Then simplify. a) 5 b) 8 c) 6 d) Example 7: Simplify. a) 4 b) 4 c) 4 d) 4 Solution: Exponents only apply to the number or letter directly below/to the left of them. The presence of parentheses indicate if an exponent is to be applied to a sign or more than one factor at a time. a) The 4 is enclosed in parentheses so the exponent applies to both the negative sign and the b) Since there are no parentheses, the exponent only applies to the Carreon 5

6 Section.5 Exponents and Order of Operations c) The 4 is enclosed in parentheses so the exponent applies to both the negative sign and the d) Since there are no parentheses, the exponent only applies to the You Try It 7: Decide whether each of the following gives a positive or negative result. Then simplify. a) b) c) d) Recall The mathematical expression inside the grouping symbols is called a quantity. Grouping symbols (parentheses, brackets, braces, and absolute value bars) act as a barrier to the quantity until the quantity is simplified down to a single number. We always perform the operations inside the grouping symbols first. When we have to simplify expressions with more than one operation, we follow the Order of Operations: Order of Operations ) Perform all operations inside the grouping symbols. Start with the innermost grouping and perform all operations inside until you are left with a single number. ) Evaluate any exponents in the expression. ) Moving from left to right, perform any multiplications or divisions in the order in which they appear. 4) Moving from left to right, perform any additions or subtractions in the order in which they appear. 05 Carreon 6

7 Section.5 Exponents and Order of Operations Example 8: Simplify. 5 7 Solution: Perform the subtraction within the parentheses: 5 7 Multiply: Add 6 You Try It 8: Simplify. 8 4 Example 9: Simplify. 6 5 Solution: Perform the subtraction within the parentheses: 5 Simplify the exponent part: 4 Divide You Try It 9: Simplify Carreon 7

8 Section.5 Exponents and Order of Operations Example 0: Simplify Solution: 48 8 Multiplications and divisions are performed in the order that they appear. 6 You Try It 0: Simplify. 0 Division appears first on the right: Multiply Example : Simplify Solution: 6 54 Perform the operations in the absolute value bars first. Multiply: You Try It : Simplify. 5 Subtract: Simplify: Carreon 8

9 Section.5 Exponents and Order of Operations Example : Simplify. 5 5 Solution: 5 5 Here, the fraction bar acts like a grouping symbol. First perform the operations in the numerator: 9 0 Then perform the operations in the denominator: Now replace original problem with simplified numerator and denominator: Divide: You Try It : Simplify Example : Simplify. 4 Solution: Simplify the exponent parts: 5 5 and Subtract. 05 Carreon 9

10 4 Section.5 Exponents and Order of Operations You Try It : Simplify. 4 Example 4: Simplify Solution: Perform the subtraction within the inner parentheses first: and Perform the multiplication within the brackets: 0 0 Subtract You Try It 4: Simplify Carreon 0