Chapter 7 - Exponents and Exponential Functions 7-1: Multiplication Properties of Exponents 7-2: Division Properties of Exponents 7-3: Rational Exponents 7-4: Scientific Notation 7-5: Exponential Functions 7-6: Growth and Decay
7-1: Multiplication Properties of Exponents Monomials are numbers, variables, or the product of numbers and variables. Constants are monomials that are numbers. Monomials: 6, x, 5m, 7x 2, xyz, x/8 Not Monomials: g + 7, 8/x, h - 2j
Whenever a term is in the form x n we refer to it as a power. The word power is often used to refer to an exponent. base 5 3 exponent The above example can be read five to the third power or five cubed.
When multiplying terms that have the same base you must add the exponents and multiply the coefficients (constants). x 3 x 4 = x 3+4 = x 7 3a 2 a 3 4a 12 = (3 4)a 2+3+12 = 12a 17 4 2 4 4 = 4 2+4 = 4 6 = 4,096
Simplify each expression. 1. (4m 3 )(m 2 ) 2. (5xy 3 )(3x 4 y) 3. (2a 2 bc 3 )(a 4 b 6 c 9 ) 4. (3k 4 m)(2k 2 m 3 )(7km 5 )
Whenever you raise a term that has an exponent by another exponent, you multiply powers. This is referred to as a power of powers. (x 4 ) 3 = x 4 3 = x 12 (5 3 ) 2 = 5 6 = 15,625
When you raise a product of a number and variables (monomial) you must raise every term within the parentheses by that power. (4x 2 ) 3 = (4) 3 (x 2 ) 3 = 64x 6 (-5a 2 b) 5 = (-5) 5 (a 2 ) 5 (b) 5 = -3,125a 10 b 5
Simplify each expression. 1. (3h 2 ) 3 (2h) 4 2. (4gh 2 ) 2 (-3g 2 h 3 ) 3 3. (3cd) 2 (c 3 d 2 ) 3 (4c 3 d 2 ) 4
We can use a combination of the three properties in this section to simplify expressions involving monomials. For a monomial to be simplified: Each variable appears only once There are no powers of powers All fractions are in simplest form There are no negative exponents
7-1 Homework P. 395-397 # 21-23, 27-33 odd, 40, 41-45 odd, 53, 68, 70, 76 (15 problems)
7-2: Division Properties of Exponents Multiplication and division are opposites. So when using division properties of exponents we do the opposite of the multiplication properties. Multiplication Division Product of Powers Add the exponents Quotient of Powers Subtract the exponents
A quotient of powers is when you are dividing two terms with the same base. When you are simplifying a quotient of powers you subtract the exponents. k 6 = k 6-4 = k 2 k 4 x 3 y 5 = x 3-1 y 5-3 = x 2 y 2 xy 3
The unwritten rule of exponents is that you raise whatever the exponent is touching. This is why the power of products rule that we learned in 7-1 works. This also applies to the power of a quotient property. (x/3) 3 = x 3 /3 3 = x 3 /27
The zero exponent property states that any nonzero number or expression that is raised to the power of zero will always equal one. 4 0 = 1 x 0 = 1 (-6x 3 y) 0 = 1 Remember, the exponent only affects what is touching. (5x) 0 = 5 0 x 0 = 1 1 = 1 5x 0 = 5 x 0 = 5 1 = 5
For an expression to be simplified you cannot have any negative exponents. The negative exponent property allows for the elimination of a negative exponent by using its reciprocal. x -2 = x -2 = 1 1 x 2 3-3 = 1 = 1 3 3 27
Review of what it means for an expression with exponents to be simplified. Each variable appears only once There are no powers of powers All fractions are in simplest form There are no negative exponents
Order of magnitude is a form of rounding. The order of magnitude is the number rounded to the nearest power of 10. Some examples of order of magnitude: 9 is close to 10 so its order of magnitude is 10 1 98 is close to 100 so its order of magnitude is 10 2 950 is close to 1000 so its order of magnitude is 10 3
7-2 Homework P. 403-405 # 19-24, 32-38 even, 42, 45, 51, 55-58, 69, 73 (19 problems)
7-3: Rational Exponents In math, the word rational stems from the word ratio. Ratios are a comparison of two numbers by division (aka fractions). Fractions can always be converted to decimal solutions. Therefore. Rational exponents are exponents that have either fractions (or decimals) as their power.
One of the most common rational exponents that we work with is the power of ½. Let s use our knowledge of powers of powers to evaluate a base with an exponent of ½. What happens if you square a power of ½? (x 1/2 ) 2 = x 1/2 x 1/2 = x 1/2 + 1/2 = x 1 = x What that example shows us is that if you square a value with a power of ½ you get the value itself.
Why does this work? A power of ½ is the same as a square root! c 1/2 = c 16 1/2 = 16 = ±4 (x 2 ) 1/2 = x 2 = x
The common mathematical operations (PEMDAS) have inverses. Addition s inverse is Subtraction. Multiplication s inverse is Division. The same is true with Exponents. The inverse of an exponent is its root. (We will discuss this further in Chapter 10)
There are other roots other than the square root. 2 3 = 2 2 2 = 8 So 8 1/3 = 8 = 2 The example above is called a cube root. For any numbers a and b, and positive integer n, if a n = b, then a is a nth root of b. If 3 4 = 3 3 3 3 = 81, then 81 (or 81 1/4 ) = 3
It might be easier to think of it like this: All exponents are numbers. All numbers can be made into fractions. The numerator tells you the power. The denominator tells you what root you are taking of the number. 3 2 = 3 2/1 = 9 4 1/2 = 4 1 = 4 = 2 27 1/3 = 27 1 = 27 = 3 4 4/2 = 4 4 = 256 = 16 This can also be proven by using the power of a power property!
Simplify: 343 625 5 243 1000 1/3 6561 1/4 32 2/5 27 4/3
Exponential equations are equations with exponents that are variables. Like with all equations that involve a variable, the goal is to solve for the variable that makes the equation true. To solve exponential equations we must get similar bases. Once we have similar bases we can set the exponential variable equal to the integer exponent. 2 x = 8 2 x = 2 3 x = 3
Solve each equation. 3 x = 27 4 x = 1024 81 x = 9 3 x+1 = 27 2 3x = 64 81 2x-3 = 9 x+3
7-3 Homework P. 410-413 # 18, 19, 22, 25-28, 30, 35, 38, 41, 44, 45, 48, 49, 52, 57, 60, 64, 71, 79, 81, 87, 97, 104 (25 problems)
7-4: Scientific Notation Scientific notation is a convenient way to write very large or small numbers. The purpose of this form is to make solving calculations easier. A number that is written in scientific notation is in the form: a x 10 n Where a is a number between 1 and 10 n is an integer (positive or negative whole #)
The standard form of writing a number is how we normally write numbers. Standard Form Scientific Notation 420,000 4.2 x 10 5 300 3.0 x 10 2 692,000,000 6.92 x 10 8 0.0047 4.7 x 10-3 0.00000005 5.0 x 10-8
To convert from standard form to scientific notation: 1. Move the decimal place until it is one place to the right of the first # 2. Count the number of places you moved the decimal 3. If you moved the decimal to the left your exponent n will be that positive number If you moved the decimal to the right your exponent n will be that negative number 4. Remove unnecessary zeros
To convert from scientific notation to standard form: 1. Look at the exponent n in scientific notation 2. If n < 0 move the decimal that many places to the left If n > 0 move the decimal that many places to the right 3. Insert any necessary zeros, decimals, and commas
You can use scientific notation to multiply or divide very large/small numbers. This is done by combining like terms. Multiply or divide the integers (a) by each other. Multiply or divide the bases of 10 n by each other. Then multiply the results of the two!
Evaluate and express the solution in both scientific and standard forms. 5.4 x 10 8 1.8 x 10 6 (5.4 1.8)(10 8 10 6 ) = (3)(10 8-6 ) = (3)(10 2 ) Scientific Notation: 3.0 x 10 2 Standard Form: 300
7-4 Homework P. 417-419 # 20, 21, 23, 24, 29, 33, 38, 46, 54, 55, 62, 63, 64, 67 (14 problems)
7-5: Exponential Functions Exponential functions are functions in the form: y = ab x ; where a is an integer other than zero In exponential functions the exponent is a variable. All exponential functions are nonlinear; this means that they do NOT make a straight line.
Exponential functions, like all functions, can be graphed using a function table. Exponential functions cannot be graphed using slope-intercept form. This is because they are not linear. If the integer (the product of ab) being raised is: > 1 then the graph opens up and to the right between 1 and 0 then the graph opens up and to the left < 0 then the graph opens downward
Exponential functions that move up and to the right are called exponential growth functions. They show how something grows over time. Exponential functions that move up and to the left are called exponential decay functions because they show how a something loses value over time.
Some examples of exponential functions look like: y = 2 x y = (½) x y = -3(4) x y = -12 x
y = 2 x Exponential growth y = (½) x Exponential Decay
y = -3(4)x
7-5 Homework P. 427-429 # 10, 15, 16, 25-30, 47, 53, 56, 59, 60 (14 problems)
7-6: Growth and Decay In the last section we saw that exponential functions can model growth or decay. These are commonly used in life to predict values of many items, including: interest, price, savings, etc.
When we discussed percent of change as it related to sales tax back in Chapter 2 we saw the base model of exponential growth. price = initial amount (1 + tax rate) p = a (1 + r) You buy a $5 notebook. The sales tax is 8%. How much do you pay? 5(1+.08) = 5(1.08) = $5.40 The same general equation is used to solve for exponential equations, except it involves an exponent.
The equation for exponential growth is: y = a(1 + r) t, where a is the initial amount r is the rate of change t is the time y is the final amount
The price of movie tickets have risen 5% each year since 2010. If the price for a ticket in 2010 was $10, write an equation for the price of a ticket in t year. How much would a ticket cost in 2017? y = a(1 + r) t Initial amount (a) = $10 Rate of increase (r) = 5% =.05 Time (t) = 7 years y = 10(1+0.05) 7 y = 10(1.05) 7 y = 10(1.41) y = $14.10
Compound interest is when interest is earned or paid based on previous balances (which have previous interest amounts included). Compound interest is one of the most commonly seen forms of exponential growth. Compound interest is calculated on a time period (t) in years.
The formula for compound interest is: y = a(1 + r/n) nt, where a is the initial amount r is the rate of change t is the time y is the final amount n is the number of times interest is compounded each year
If Jordan invests $1,000 at a rate of 3% per year compounded quarterly, how much money will he have after 12 years? y = a(1 + r/n) nt Initial amount (a) = 1,000 Rate (r) = 3% =.03 Number of compounded times (n) = 4 [there are 4 quarters in a whole] Time in years (t) = 12 y = 1,000(1 +.03/4) 4(12) y = 1,000(1 +.0075) 48 y = 1,000(1.0075) 48 y = 1,000(1.43) y = $1,430
Exponential decay also occurs in life. Things break down, lose their air, or lose their value as they grow older. The formula for exponential decay is nearly identical to exponential growth except that it involves subtraction within the parentheses instead of addition. The equation for exponential decay is: y = a(1 - r) t
When you buy a new car its value begins to decrease as soon as you start to drive it. Kathy bought a car for $25,500. If her car depreciates (decreases in value) at a rate of 11% per year, how much would her car be worth in 5 years? Initial amount (a) = $25,500 Rate of decrease (r) = 11% = 0.11 Time (t) = 5 years y = a(1 - r) t y = 25,500(1-0.11) 5 y = 25,500(0.89) 5 y = 25,500(0.56) y = $14,280
7-6 Homework P. 434-436 # 4-7, 12, 21, 22, 35, 39 (9 problems)