8-1 Zero and Negative Exponents
Anything to the zero power is 1. This comes from the rules for dividing exponents.
When multiplying like bases, you add the exponents, so to “undo” that process, you would be dividing and subtracting. In division, when you have the same thing on the top and the bottom, it simplifies to one.
Example: x4 * x5 = x4+5 = x9
To divide: x6/x4 = x6-4 = x2
Let’s look at x7/x7 = x7-7 = x0 = 1 since x7/x7 simplifies to 1.
A negative exponent is an inverse. It ultimately means “one over” so x-3 means one over x3 or 1/x3. We look at these negative exponents as “bad attitudes” and we need to move the base to make it happy, or positive.
It is VERY IMPORTANT to only move the base and exponent that is negative, and nothing more.
8-2 Scientific Notation
Scientific notation is a shortcut for writing extremely large and extremely small numbers. It is written as the product of two factors: a number between 1 and 10, and a power of 10.
To put a number into scientific notation, you move the decimal to the place between the first and second non-zero number. The power of ten becomes the number of places you moved the decimal – positive exponent for numbers originally greater than 10 and a negative exponent for numbers originally less than 1.
Example: 45,930,000,000,000 would become 4.593 x 1013
0.000632 would become 6.32 x 10 -4
To work with numbers in scientific notation, it may be easier to take the numbers out first, then put them back in, but you don’t have to if you learn the shortcuts.
Example: Multiply and simplify in scientific notation
5(3.4 x 105) – this initially looks like a distributive type problem,
but notice there is no addition or subtraction so it is really just a
multiplication problem. It can be compared to 3(3.4 x 10) which is
just 3 x 3.4 x 10
So to simplify: 5 x 3.4 x 105 = 17 x 105
But that is not in scientific notation since 17 is greater than 10, so
we have to adjust it. The decimal needs to move left, which means
the exponent needs to move right (or increase)
The answer then in scientific notation is 1.7 x 106
8-3 Multiplication Properties of Exponents
This section should be review. When you multiply like bases, you add exponents. Now that you know negative exponents are moved to make them positive, you can set up fractions and cancel them out, OR you can keep your exponents all on one level and have a possible negative exponent in your answer. A simplified answer will not have negative exponents!
Example: x7y4z * x-5y2z-5
8-4 More Multiplication Properties of Exponents
The shortcut rule for a power to a power is to reverse distribute the exponent.
Since the exponent tells you how many times the base is used as a factor, the base would be multiplied that exponent number of times. That in turn means the exponents would be added that number of times, which results in multiplication. (Recall from elementary school that multiplication is a short cut for long addition by the same number.)
Example: (3xy4z2)3 = 31*3x1*3y4*3z2*3 = 33x3y12x6 = 27 x3y12x6
Remember everything has an exponent, so if you don’t see one there, it is automatically a one!
8-5 Division Properties of Exponents
This sections should also be review – to divide when you have the same base, you subtract the exponents. Keep in mind now that you may be subtracting a negative number!
Example: x4y-2z/x-3y5z6 = x4- -3y-2-5z1-6 = x7y-7z-5 = x7/y7z5
The other option is to move the variables first to get positive exponents, then cancel out.
A quotient to a power will still “reverse distribute” to everything in parenthesis.
Example: (x2/y3)2 = x2*2/y3*2 = x4/y6
8-6 Geometric Sequences
We looked at Arithmetic sequences a while back, and we discussed geometric sequences at that time.
Recall that an Arithmetic sequence has a common difference, or something that is added or subtracted in the sequence to get the next term.
A geometric sequence has a common ratio, or something that is multiplied or divided each time in the sequence to get the next term.
Example: 3, 5, 7, 9, 11… would be arithmetic since 2 is added each time to get the next term. 2 is called the common difference
3, 6, 12, 24, 48…would be a geometric sequence since 2 is multiplied each time to get the next term. 2 is the common ratio.
Just like the Arithmetic sequence had a Rule –
A(n) = ao + (n-1)d where ao is the initial value of the sequence, d is the common difference and n is the term number you are looking for.
So does the Geometric Sequence have a Rule –
A(n) = ao* rn-1 where ao is the initial value of the sequence, r is the common ratio and n is the term number you are looking for.
Example: Find the 16th term of this sequence: -24, 8, -8/3, 8/9…
First, find the Common Ratio:
Then use the Rule to determine the 16th term:
8-7 Exponential Functions
Exponential Functions are functions whose exponent is a variable. The graph of this type of equation is generally a curve. Since anything to the zero power is 1, the parent function should always have a y intercept of 1 (when x = 0) assuming there is no shift.
If the leading coefficient is negative, that will flip the graph to under the y-axis. If the initial value (a) is less than one but positive, the graph will be getting smaller, if it is greater than one it will be getting bigger.
Sketch a rough graph:
Y = (1/3)x y = 3x
Y = - (1/3)x y = -3x
The general form for an Exponential Function is y = a * bx where a is a constant (and the beginning value) b is the base and x the exponent
8-8 Exponential Growth and Decay
Exponential functions are used in the real world when dealing with growth and decay. This can range from the interest on a bank account earning compound interest to the rate of radioactive decay used in Carbon dating.
Using the general form, y = a * bx a is the starting amount (such as initial deposit in a bank account) and b is one plus/minus the rate (often called the growth factor or decay factor) and the exponent is a period of time.
Example: Suppose you deposit $1200 into an account paying 5.5% interest compounded semi-annually. How much money will you have after 3 years?
a = 1200 (the initial deposit)
This is a growth model so
b = 1 + rate = 1 + .055 = 1.055
x = the number of interest periods – in 3 years, there will be 6 semi-annual interest periods.
So for y = a * bx we now have y = (1200)*(1.055)6 = $1654.61
Anything to the zero power is 1. This comes from the rules for dividing exponents.
When multiplying like bases, you add the exponents, so to “undo” that process, you would be dividing and subtracting. In division, when you have the same thing on the top and the bottom, it simplifies to one.
Example: x4 * x5 = x4+5 = x9
To divide: x6/x4 = x6-4 = x2
Let’s look at x7/x7 = x7-7 = x0 = 1 since x7/x7 simplifies to 1.
A negative exponent is an inverse. It ultimately means “one over” so x-3 means one over x3 or 1/x3. We look at these negative exponents as “bad attitudes” and we need to move the base to make it happy, or positive.
It is VERY IMPORTANT to only move the base and exponent that is negative, and nothing more.
8-2 Scientific Notation
Scientific notation is a shortcut for writing extremely large and extremely small numbers. It is written as the product of two factors: a number between 1 and 10, and a power of 10.
To put a number into scientific notation, you move the decimal to the place between the first and second non-zero number. The power of ten becomes the number of places you moved the decimal – positive exponent for numbers originally greater than 10 and a negative exponent for numbers originally less than 1.
Example: 45,930,000,000,000 would become 4.593 x 1013
0.000632 would become 6.32 x 10 -4
To work with numbers in scientific notation, it may be easier to take the numbers out first, then put them back in, but you don’t have to if you learn the shortcuts.
Example: Multiply and simplify in scientific notation
5(3.4 x 105) – this initially looks like a distributive type problem,
but notice there is no addition or subtraction so it is really just a
multiplication problem. It can be compared to 3(3.4 x 10) which is
just 3 x 3.4 x 10
So to simplify: 5 x 3.4 x 105 = 17 x 105
But that is not in scientific notation since 17 is greater than 10, so
we have to adjust it. The decimal needs to move left, which means
the exponent needs to move right (or increase)
The answer then in scientific notation is 1.7 x 106
8-3 Multiplication Properties of Exponents
This section should be review. When you multiply like bases, you add exponents. Now that you know negative exponents are moved to make them positive, you can set up fractions and cancel them out, OR you can keep your exponents all on one level and have a possible negative exponent in your answer. A simplified answer will not have negative exponents!
Example: x7y4z * x-5y2z-5
8-4 More Multiplication Properties of Exponents
The shortcut rule for a power to a power is to reverse distribute the exponent.
Since the exponent tells you how many times the base is used as a factor, the base would be multiplied that exponent number of times. That in turn means the exponents would be added that number of times, which results in multiplication. (Recall from elementary school that multiplication is a short cut for long addition by the same number.)
Example: (3xy4z2)3 = 31*3x1*3y4*3z2*3 = 33x3y12x6 = 27 x3y12x6
Remember everything has an exponent, so if you don’t see one there, it is automatically a one!
8-5 Division Properties of Exponents
This sections should also be review – to divide when you have the same base, you subtract the exponents. Keep in mind now that you may be subtracting a negative number!
Example: x4y-2z/x-3y5z6 = x4- -3y-2-5z1-6 = x7y-7z-5 = x7/y7z5
The other option is to move the variables first to get positive exponents, then cancel out.
A quotient to a power will still “reverse distribute” to everything in parenthesis.
Example: (x2/y3)2 = x2*2/y3*2 = x4/y6
8-6 Geometric Sequences
We looked at Arithmetic sequences a while back, and we discussed geometric sequences at that time.
Recall that an Arithmetic sequence has a common difference, or something that is added or subtracted in the sequence to get the next term.
A geometric sequence has a common ratio, or something that is multiplied or divided each time in the sequence to get the next term.
Example: 3, 5, 7, 9, 11… would be arithmetic since 2 is added each time to get the next term. 2 is called the common difference
3, 6, 12, 24, 48…would be a geometric sequence since 2 is multiplied each time to get the next term. 2 is the common ratio.
Just like the Arithmetic sequence had a Rule –
A(n) = ao + (n-1)d where ao is the initial value of the sequence, d is the common difference and n is the term number you are looking for.
So does the Geometric Sequence have a Rule –
A(n) = ao* rn-1 where ao is the initial value of the sequence, r is the common ratio and n is the term number you are looking for.
Example: Find the 16th term of this sequence: -24, 8, -8/3, 8/9…
First, find the Common Ratio:
Then use the Rule to determine the 16th term:
8-7 Exponential Functions
Exponential Functions are functions whose exponent is a variable. The graph of this type of equation is generally a curve. Since anything to the zero power is 1, the parent function should always have a y intercept of 1 (when x = 0) assuming there is no shift.
If the leading coefficient is negative, that will flip the graph to under the y-axis. If the initial value (a) is less than one but positive, the graph will be getting smaller, if it is greater than one it will be getting bigger.
Sketch a rough graph:
Y = (1/3)x y = 3x
Y = - (1/3)x y = -3x
The general form for an Exponential Function is y = a * bx where a is a constant (and the beginning value) b is the base and x the exponent
8-8 Exponential Growth and Decay
Exponential functions are used in the real world when dealing with growth and decay. This can range from the interest on a bank account earning compound interest to the rate of radioactive decay used in Carbon dating.
Using the general form, y = a * bx a is the starting amount (such as initial deposit in a bank account) and b is one plus/minus the rate (often called the growth factor or decay factor) and the exponent is a period of time.
Example: Suppose you deposit $1200 into an account paying 5.5% interest compounded semi-annually. How much money will you have after 3 years?
a = 1200 (the initial deposit)
This is a growth model so
b = 1 + rate = 1 + .055 = 1.055
x = the number of interest periods – in 3 years, there will be 6 semi-annual interest periods.
So for y = a * bx we now have y = (1200)*(1.055)6 = $1654.61