1. R ATIONAL E XPONENTS AND M ORE W ORD P ROBLEMS

2. R EVIEW Last class we saw that Powers with a common base can be easily multiplied and divided (2x 5 )(3x 3 ) = 6x 8 2 8 / 2 5 = 2 3 = 8 (2x 5 )(3x 3 ) = 6x 8 2 8 / 2 5 = 2 3 = 8 Powers themselves can be raised to another power (4d 6 ) 2 = 16d 12 Powers with a negative exponent are equivalent to the reciprocal base and positive exponent. 3e -3 = 3 e 3 3e -3 = 3 e 3 Exponential equations can be solved if the powers contain the same base. 4 2x = 8 3 (2 2 ) 2x = (2 3 ) 3 2 4x = 2 9 4x = 9 x = 9/4 4 2x = 8 3 (2 2 ) 2x = (2 3 ) 3 2 4x = 2 9 4x = 9 x = 9/4

3. R ATIONAL EXPONENTS A rational number is one that can be expressed as a fraction a. b The numbers π and √2 are examples of irrational numbers. Ex. Solve for x. We know to solve this we must “undo” the square by taking the square root of both sides. But what exactly is the square root? In order for the square root to undo the exponent 2, it must also be an exponent. In order for the square root to undo the exponent 2, it must also be an exponent. Notice that in the final answer the ‘x’ term has an exponent of 1. 1 So the problem is actually solved like this…

4. R ATIONAL EXPONENTS Ex. Solve for x. We raise both sides to an exponent. But what is this exponent? We see that one the left-hand side we have a power of a power so we will multiply the exponents to get 1. 1 (2)(?) = 1 The missing exponent is 1/2 !! So the square root is actually an exponent of 1/2

5. R ATIONAL E XPONENTS So what does an exponent of 1/3 mean? Ex. Solve for x 8 1/3 = x We could cube both sides. The left-hand side is a power of a power so we multiply the exponents. We know that 2 3 is 8 so x = 2. This means that an exponent of 1/3 is the same as the cubed- root. It is like asking what cubed gives me this base? In fact all the rational exponents are like this. Ex. a) 64 1/6 b) 81 1/4 c. 125 1/3 a) 2 (2 6 = 64)b) 3 (3 4 = 81)c. 5 (5 3 = 125)

6. R ATIONAL E XPONENTS Remember that this all started by recognizing that the square root was an exponent. It follows, then, that all rational exponents can be represented by root signs. Ex. In all of these examples the bases are perfect squares. We can write them as so.

7. R ATIONAL E XPONENTS In each case we have a power of a power and can multiply the exponents. So now we can have rational exponents where the numerator is not 1.

8. R ATIONAL E XPONENTS Evaluate the following with a calculator. Show one intermediate step. When we need to evaluate an expression with a rational exponent, we can “rip the exponent apart”. The exponent can be viewed as a square and an exponent of 1/3 However, we are not responsible to know 125 2. Is there another way? Since the new expression is a power of a power, we multiply the exponents. It does not matter what order there are in. So let’s switch them. We know that 125 1/3 is 5!

9. R ATIONAL E XPONENTS Evaluate the following with a calculator. Show one intermediate step. When we need to evaluate an expression with a rational exponent, we can “rip the exponent apart”. The exponent can be viewed as a cube and an exponent of 1/2 OR… The exponent can be viewed as an exponent of ½ and a cube And we know the square root of 9 and 4.

10. R ATIONAL E XPONENTS Try these: 1. Evaluate each without a calculator by showing an intermediate step. a) 4 5/2 b) 81 -3/4 c) 100 -2.5 a) b) c)

11. R ATIONAL E XPONENTS Try these: 1. Evaluate each without a calculator by showing an intermediate step. d) e) f) d) e)f)

12. R ATIONAL E XPONENTS Try these: 2. Solve the exponential equation Notice that we do not know the cubed-root of 32 or the square root of 8. But we can get COMMON BASES Now simplify each side, remembering that a square root is an exponent of ½ and the cubed-root is the exponent 1/3 Power of a power…. Since the bases are equal so are the exponents.

13. R ATIONAL E XPONENTS We can (finally) solve for x. 3. Solve for x.

14. B ACK TO S OME W ORD P ROBLEMS We have seen that patterns with a common ratio can be described with an exponential equation. Ex. 120,60,30,15,7.5… This can be written as We have also seen that when a data table is given we can use this general equation: Where ‘a’ is the initial amount (at the beginning) and period is when the value of y increases by r times Where ‘a’ is the initial amount (at the beginning) and period is when the value of y increases by r times

15. M ORE W ORD P ROBLEMS Where ‘a’ is the initial amount (at the beginning) and period is when the value of y increases by r times Ex 1. A certain population has been seen to triple every 12 years. In 1950, there was 2500 individuals. How many was there in 2011? The equation is: We need to solve for y knowing that x = 2011-1950 In 2011 there was 665718 individuals.

16. M ORE W ORD P ROBLEMS Where ‘a’ is the initial amount (at the beginning) and period is when the value of y increases by r times Ex 2. A certain population has been seen to triple every 12 years. In 1950, there was 2500 individuals. When will there be 67500 individuals? The equation is: We need to solve for x knowing that y = 67500 In 36 years (1986) there would be 67500 individuals. Isolate the power!! Common bases anyone?

17. M ORE W ORD P ROBLEMS Ex 3. A teacher’s salary increases by 6% every 15 years. In 2000, a starting teacher’s salary was $35000. What will it be in 2020? We can start this by looking at a table of values. x (years since 2000)0153045 Starting salary (in thousands of dollars) 35 To calculate the salary after 15 years, we can take 6% of the present salary and add it to the present salary. 6% of 35 = 0.06(35) = 2.1 2.1 + 35 = 37.1 37.1 Now take 6% of 37.1 and add it to 37.1 39.33 Now take 6% of 39.33 and add it to 39.33 41.69 I’m rich!

18. M ORE W ORD P ROBLEMS Ex 3. A teacher’s salary increases by 6% every 15 years. In 2000, a starting teacher’s salary was $35000. What will it be in 2020? We can start this by looking at a table of values. x (years since 2000)0153045 Starting salary (in thousands of dollars) 35 37.139.3341.69 Now we can find the equation. So let’s find the common ratio. 1.06 1.06 1.06 Of course! We got each successive value by taking 6% of the last value and adding it to the last value. 6% of x = 0.06x 0.06x + x = 1.06x Each term is 1.06 times as big as the last

19. M ORE W ORD P ROBLEMS Ex 3. A teacher’s salary increases by 6% every 15 years. In 2000, a starting teacher’s salary was $35000. What will it be in 2020? We can start this by looking at a table of values. x (years since 2000)0153045 Starting salary (in thousands of dollars) 35 37.139.3341.69 When a value increases or appreciates by a certain rate, the common ratio will be 1 + rate/100 When a value decreases or depreciates by a certain rate, the common ratio will be 1 - rate/100

20. M ORE W ORD P ROBLEMS Ex 3. A teacher’s salary increases by 6% every 15 years. In 2000, a starting teacher’s salary was $35000. What will it be in 2020? We can start this by looking at a table of values. x (years since 2000)0153045 Starting salary (in thousands of dollars) 35 37.139.3341.69 So the equation for this data is Plug in 20 for x (years since 2000) to solve for y In 2020 a starting teacher will make $37 830

21. L I OR F E Another exponential situation involves radioactive material. Certain isotopes of the elements that make up the world around us are unstable. Each nucleus decays randomly but overall there is a pattern. We model this with HALF-LIFE Ex. A certain radioactive substance has a half of 8 days. It initially contained 90 mg. When will there be 5.625 mg left? The half life of a substance is the time for half of the material to decay into something else. After two half-lives, ¼ of the original amount remains. All half-life questions have a common ratio (and base) of ½. In 32 days, there will be 5.625 mg left.

22. L I OR F E Ex 2. A certain substance has a half-life of 65 minutes. When will there be of the original amount? Notice that we were not given the original amount. Notice that we were not given the original amount. However, we can still solve the problem. The equation would be The next step would be to divide both sides by ‘a’. The y value is 1/64 times ‘a’ or

23. I NTEREST I NTEREST Another type of exponential equation is the compound interest equation. This is different than simple interest which simply gives you a set amount each time. Compound interest gives you a set percentage of the amount in your account. This amount may include some interest already earned. With compound interest you’re getting interest on the interest. Ex. You invest $100 in an account paying 6% interest a year compounded quarterly. How much will you have in 10 years? You get 6% in the year but you get it spread over 4 times (quarterly). Each time you get an interest payment you’re getting 6%/4 = 1.5%

24. I NTEREST Ex. You invest $100 in an account paying 6% interest a year compounded quarterly. How much will you have in 10 years? So the equation would look like this: The amount of interest earned each period x is time in years. The period is ¼ of a year Simplifying the exponent gives Now we can plug in x =10 y = $181.40

25. I NTEREST In general, the compound interest is Where  A is the amount in the account at time, t  P is the principle (initial) amount  r is the decimal value of the interest rate  n is how many times per year the interest is compounded. Look for terms like: daily (n =365), semi-annually (n = 2), weekly (n = 52) and monthly (n =12)

26. I NTEREST Ex 2. An bank account earns interest compounded monthly. The investment doubles in 9.27 years. Calculate the annual interest rate. When the money doubles there will be 2P in the account. So A = 2P We are now solving for the base. We must “undo” the exponent. Raise both sides to the 1/111.25.

27. I NTEREST Ex 2. An bank account earns interest compounded monthly. The investment doubles in 9.27 years. Calculate the annual interest rate.

28. I NTEREST Ex 3. Which is better: 5% interest per year compounded monthly or 5% per year compounded daily? Let’s assume an initial investment of $100 and a term of 10 years. Because the interest is compounded more often (even though each time it is a smaller percentage) the account paying the daily compounded interest is better.