Operations with Whole Numbers and Exponents. 30A Positive exponents Let s apply the product and quotient rules for exponents to solve a few examples.

Transcription

1 Lesson 30 Operations with Whole Numbers and Exponents Review: Lesson 3, rules for Lesson 7 applied to numbers. Lessons will be related to operations with numbers and numerical bases. Then, in Lessons we will transition from arithmetic to algebra, replacing numbers with letters. The concepts you will learn over the next 0 lessons are some of the foundations of algebra, so if you can learn these, the rest of algebra should be easy, right? Right! 30A Positive exponents Let s apply the product and quotient rules for exponents to solve a few examples. Example 30. Simplify to eliminate all unnecessary exponents. 6 2 i 5 2 i solution: Simplify to eliminate all unnecessary exponents means that you don t need to calculate a single numerical answer. Just use the product and quotient rules for exponents, as needed, to simplify. Both the product and quotient rule can be used together, because all the quotient rule does is turn a division problem into a multiplication problem. Also, from Lesson 3 we learned about the commutative property, which tells us that in multiplication, order doesn t matter, so we can rearrange the order of factors, putting similar bases next to each other. 6 2 i 5 2 i 6 3 = 6 2 i 6 3 i 5 2 i 5 = i 5 2 = 6 5 i 5 Example 30.2 Simplify to eliminate all unnecessary exponents. i 7 4 i 6 3 solution: As in Example 30., turn it into a multiplication problem first, then simplify. i 7 4 i 6 3 = i 7 4 i 7 5 i 6 3 = +4 5 i 6 3 = 7 4 i 6 3

2 Alternatively, if you recognized that the terms in the numerator and denominator would cancel, you could have simplified by crossing them out first: i 7 4 i i = 7 0, and anything (except 0) raised to the power of 0 equals (Lesson 7), which is why over cancels. It just equals. Example 30.3 Simplify to eliminate all unnecessary exponents solution: Just like regular arithmetic problems, the first step should be to eliminate parentheses. Next, turn it into two multiplication problems, separated by a plus sign, and simplify = 2 3 i i i i 2 = i i = 2 2 i i 3 3 Instead of 2, it is also okay just to write 2. 30B Negative Numbers and Negative Exponents Some of the most frequent mistakes students make in algebra involve negative signs. Students either forget them or misuse them. You may find yourself referring back to this lesson frequently until you have fully grasped working with negative exponents and negative numbers with exponents. Example 30.4 Simplify. Write answer as a number without exponents. a) -4 2 b) -4-2 c) (-4) 2 d) (-4) -2 solution: A) Recall from Lesson 3 that in algebra, it is helpful to think of the negative sign as the opposite of. So, we might say this example to ourselves as the opposite of 4 squared, which is the opposite of 6. Therefore, the answer is -6. B) We would say this problem to ourselves as the opposite of over 4 squared, which

3 is the opposite of over 6. Therefore, the answer is 6. C) The parentheses around the negative 4 are incredibly important. They group the negative with the 4, as opposed to making the whole value negative as in Examples A and B. You would say this problem as the opposite of 4 times the opposite of 4, or (-4) (-4)=6. D) This is the same as ( 4) 2, and since (-4)2 = 6, Example D equals 6. Recapping, look again at what each example simplifies to, paying careful attention to where the negative sign is located. Is it describing the entire value as negative, or just an individual number within the expression? a) -4 2 = -6 b) -4-2 = 6 c) (-4) 2 = 6 d) (-4) -2 = 6 Example 30.5 Simplify to eliminate all unnecessary exponents. i 7 4 i 6 3 solution: This is similar to Example 30.2, with the exception that the answer contains negative exponents. i 7 4 i 6 3 = i 7 4 i 6 3 i 6 5 = i = 7 i 6 2 Example 30.6 Simplify to eliminate all unnecessary exponents solution: This is similar to Example 30.3, with the exception that the answer contains negative exponents = 2 3 i i i i 2 = i i i i 3 3

4 Practice Set 30 Use your best judgement as to when you should and shouldn t use a calculator. 30. Simplify to eliminate all unnecessary exponents. 8 2 i 5 2 i Simplify to eliminate all fractions and all unnecessary exponents Simplify. Write answer as a number without exponents Simplify. Write answer as a number without exponents. (-3) Simplify. Write answer as a number without exponents. -(-3) Simplify to eliminate all unnecessary exponents. 8 2 i 5 4 i Simplify to eliminate all fractions and all unnecessary exponents (SAT) A sports equipment website in Switzerland receives many orders from U.S. customers. When an American makes a purchase with a BankWorld credit card, BankWorld converts the purchase price from euros to U.S. dollars (USD), and then charges a fee on the converted cost. Katie bought a pair of soccer cleats for $00 euros. If the exchange rate was 0.76 euros per USD, calculate the fee as a percentage if the the total charged to Katie s card equaled $39.48 USD. Round answer to nearest whole number Fiesta Restaurant Group s ratio of annual revenue to net income was 60:0. If their net annual income equaled 9 million dollars, what was their annual revenue? The map scale was in 63,000. If the two cities were 2.7 inches apart on the map, what was the actual distance, in inches, between them? 27. On Wednesday, the Dow Jones Industrial Average closed times higher than Tuesday s closing. If Tuesday s average was 6,000, what was Wednesday s average? If the Nasdaq Composite Index closed at 4237 Thursday and 486 Wednesday, how many times greater was Thursday s index? Round answer to 3 dec. places Ruth ordered a pork rib plate for $2.99 and a medium drink for $.29. Her mother ordered the cobb salad for $.99. How much more did Ruth s food cost than her mother s?

5 4 26. James had $0,000 to invest in the stock market. Company A s stock was $50 per share, and Company B s stock was $60 per share. If James bought 00 shares of company A s stock, what is the maximum number of shares of Company B s stock he could purchase with his remaining money? Find he common ratio for the following sequence:, 2, 33, 464, The bag contained 2 white marbles and 3 blue marbles. What is the probability of reaching in the bag, without looking, and removing a white marble? Find the range for the temperature data from Problem Here is the data: 69, 65, 65, 66, 64, 62, 6, 65, 65, Find the perimeter of the following shape. It may not look like it at first glance, but all the values needed to calculate the perimeter are shown. Dimensions are in feet Simplify. a3 z 3 z Simplify 49