Exponents and Order of Operations

Exponents and Order of Operations Use exponents to write repeated factors. Simplify expressions containing exponents. Use the order of operations. Simplify expressions with fraction bars.

Exponents and Order of Operations An exponent is a quick way to write repeated multiplication. For example, Exponent 3 • 3 • 3 • 3 can be written 34 Base This is called exponential notation or exponential form. To simplify 34, actually do the multiplication. 34 = 3 • 3 • 3 • 3 = 81

Reading Common Exponents Here are some examples of how to read common exponents. 31 is read “3 to the first power.” 32 is read “3 to the second power” or, more commonly, “3 squared.” 33 is read “3 to the third power” or, more commonly, “3 cubed.” 34 is read “3 to the fourth power.” 35 is read “3 to the fifth power.”

Exponents and Order of Operations EXAMPLE 1 Using Exponents Rewrite each multiplication using exponents. Also indicate how to read the exponential form. (a) 10 • 10 • 10 • 10 • 10 • 10 can be written as 106, which is read “10 to the sixth power.” (b) ( 7 )( 7 ) can be written as 72, which is read “7 squared” or “7 to the second power.” (c) 2 can be written as 21, which is read “2 to the first power.”

Exponents and Order of Operations EXAMPLE 2 Using Exponents with Negative Numbers Simplify. (a) ( –2 )3 = ( –2 ) ( –2 ) ( –2 ) 4 ( –2 ) –8 (b) ( –2 )4 = ( –2 ) ( –2 ) ( –2 ) ( –2 ) –8 ( –2 ) From 2(a), ( –2 )3 = –8. 16

Exponents and Order of Operations EXAMPLE 2 Using Exponents with Negative Numbers Simplify. ( –5 )2 ( –2 )3 (c) ( –5 )2 ( –2 )3 = ( –5 ) ( –5 ) ( –2 ) ( –2 ) ( –2 ) ( 25 ) ( –8 ) –200

Exponents and Order of Operations Calculator Tip On your TI-30X IIS calculator, use the exponent key to enter exponents. To enter 75, press the following keys. ^ 7 ^ 5 = 16807 ^

Exponents and Order of Operations Calculator Tip On your TI-30Xa calculator, use the exponent key to enter exponents. To enter 75, press the following keys. yx 7 yx 5 = 16807 yx

Exponents and Order of Operations EXAMPLE 3 Working from Left to Right Simplify. (a) –3 – –8 + –2 Do additions and subtractions from left to right. 5 + –2 3

Exponents and Order of Operations EXAMPLE 3 Working from Left to Right Simplify. (b) –20 ÷ 2 • 5 Do multiplications and divisions from left to right. –10 • 5 –50

Exponents and Order of Operations Compare the methods used to simplify the following example. 10 + 2 • 3 If we work from left to right If we multiply first 10 + 2 • 3 10 + 2 • 3 12 • 3 10 + 6 36 16 Mathematicians have agreed to do things in a certain order. In this example, we multiply before we add.

Exponents and Order of Operations Step 1 Work inside parentheses or other grouping symbols. Step 2 Simplify expressions with exponents. Step 3 Do the remaining multiplications and divisions as they occur from left to right. Step 4 Do the remaining additions and subtractions as they occur from left to right.

Exponents and Order of Operations CAUTION To help in remembering the order of operations, you may have memorized the letters PEMDAS, or the phrase “Please Excuse My Dear Aunt Sally.” Please Excuse My Dear Aunt Sally 1. Parentheses 2. Exponents 3. Multiply & Divide (from left to right) 4. Add & Subtract (from left to right) Be careful! Do not automatically do all multiplication before division. Multiplication and division are done from left to right. Likewise, addition and subtraction are done from left to right.

Exponents and Order of Operations Calculator Tip Enter the previous example in your calculator. 10 + 2 x 3 = 16 If you have a scientific calculator, it automatically uses the order of operations and multiplies first to get the correct answer of 16.

Exponents and Order of Operations EXAMPLE 4 Using the Order of Operations with Whole #’s Simplify. 5 + 2 ( 24 – 4 • 2 ) ÷ 8 Multiply inside parentheses first. 5 + 2 ( 24 – 8 ) ÷ 8 Subtract inside parentheses. 5 + 2 ( 16 ) ÷ 8 Multiply. 5 + 32 ÷ 8 Divide. 5 + 4 Add. 9

Exponents and Order of Operations EXAMPLE 5 Using the Order of Operations with Integers Simplify. (a) –40 ÷ ( 12 – 7 ) – 4 Subtract inside parentheses first. –40 ÷ 5 – 4 Divide. –8 – 4 Subtract. –12

Using the Order of Operations with Integers EXAMPLE 5 Using the Order of Operations with Integers Simplify. (b) 6 + 5 ( 1 – –3 ) • ( 18 ÷ –9 ) Subtract inside parentheses first. 6 + 5 ( 4 ) • ( 18 ÷ –9 ) Divide inside parentheses. 6 + 5 ( 4 ) • ( –2 ) Multiply from left to right. 6 + 20 • ( –2 ) Multiply. 6 + –40 Add. –34

Exponents and Order of Operations EXAMPLE 6 Using the Order of Operations with Exponents Simplify. (a) ( –5 )2 – ( –2 )3 Apply exponents. 25 – –8 Add. 33

Exponents and Order of Operations EXAMPLE 6 Using the Order of Operations with Exponents Simplify. (b) ( –3 )3 – ( 8 – 6 )3 ( 3 )2 Subtract inside parentheses first. ( –3 )3 – ( 2 )3 ( 3 )2 Apply exponents. –27 – 8 ( 9 ) Multiply. –27 – 72 Add. –99

Exponents and Order of Operations EXAMPLE 7 Using the Order of Operations with Fraction Bars –4 – 2 ( 3 – –1 )2 3 • –4 ÷ 2 • –3 –36 18 Simplify. = = –2 First, do the work in the numerator Next, do the work in the denominator –4 – 2 ( 3 – –1 )2 3 • –4 ÷ 2 • –3 –4 – 2 ( 4 )2 –12 ÷ 2 • –3 –4 – 2 ( 16 ) –6 • –3 –4 – 32 18 –36