High School Algebra I Unlocked (2016)
Chapter 2. Exponents and Sequences
GOALS
By the end of this chapter, you will be able to:
•Simplify exponents using the rules of exponents
•Manipulate and work with negative exponents
•Explain the special rules of exponents
•Complete the prime factorization of large numbers
•Solve expressions with non-integer exponents
•Identify equivalent exponential expressions
•Identify patterns of sequences, series, and summations
•Solve both arithmetic and geometric sequences
Lesson 2.1. Exponents
REVIEW
BEFORE BEGINNING THIS CHAPTER, YOU SHOULD BE FAMILIAR WITH:
the definitions of rational number and irrational number
number-based exponents, such as 3^{2} = 9
number-based radicals, such as = 2
the definition and concept of prime number
the definition and concept of factorization
working with number sequences
Exponents are used to simplify numeric expressions. While exponents can work with any number, in this book we’ll deal only with rational numbers. A rational number is a number that can be written as a ratio, or fraction, or two numbers. When describing the capacity of a device, for example, it is far easier to say that a computer has 50 × 10^{9} gigabytes than it is to say a computer has 50 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 gigabytes.
Exponent Basics
The exponent associated with a base number tells you how many times you should multiply the base number by itself. Consider the following:
14^{3} = 14 × 14 × 14
14 × 14 × 14 = 2,744
In this example, the base number is 14, and the exponent, also referred to as the power, is 3. This means that the value of 14^{3} can be found by multiplying the base number by itself three times.
Likewise, if you are given a variable rather than a number, you simply multiply the variable by itself the indicated number of times.
x^{2} = x × x a^{2}b^{3} = a × a × b × b × b
Here, x^{2} indicates that x, the base, has a power of 2 and thus must be multiplied by itself twice: x^{2} = x × x. Similarly, in the expression a^{2}b^{3}, a is multiplied by itself twice, and b is multiplied by itself three times: a^{2}b^{3} = a × a × b × b × b.
You can also have negative exponents. When working with negative exponents, the exponent indicates how many times to divide 1 by the number. Take a look at the following:
14^{−3} = 1 ÷ 14 ÷ 14 ÷ 14
= 0.00036443
Any number to the
power of zero is 1:
1^{0} = 1, 4^{0} = 1, 1298^{0} = 1
But what about 0^{0}, you ask?
Mathematicians can’t
agree upon the value of 0^{0}.
Some argue that 0^{0} = 0,
while others argue that
0^{0} = 1. Weird, right?
You can find the value of a negative exponent by first converting to a positive exponent, and then taking the reciprocal of the positive exponent. A reciprocal of a given number is found by dividing 1 by that number. For example, the reciprocals of 2, −4, and 1/2 are 1/2, −1/4, and 2, respectively.
Imagine that you wanted to represent 14^{−3} as a positive exponent and then as a positive number. You could do the following:
To change the sign of an
exponent from negative
to positive, or positive to
negative, use a reciprocal!
As before, the value of 14^{−3} is 0.00036443. Regardless of the method you choose when working with negative exponents, you will arrive at the same answer.
You may also be asked to write an expression using only positive exponents. Use the rules of exponents to convert negative exponents to positive exponents:
On the other hand, if you are given an expression that has a positive exponent in the denominator, you should recognize that, in non-fraction form, the expression has a negative exponent:
You should also be able to convert large numbers to exponents. In order to do this, you need to find the prime factors of a number. What’s a prime factor, you ask? Let’s create a prime factor tree for the numbers 27, 36, and 121 and walk through what these prime factor trees are illustrating.
On a prime factor tree, you have the ancestor, parents, children, and leaves. The ancestor of the tree is the initial number; in the first tree above, the ancestor is 27. 27 would also be considered a parent, as it has two “children.” In the figure on the left, the smallest prime factor of 27 is three, which goes into 27 nine times, indicating that the children are 3 and 9. (Note that the product of the two children will always equal the parent.)
You’re now concerned with 9, the only child that can be further factored. The prime factors of 9 are 3 and 3, so 9 has two children, both of which are leaves. Leaves are numbers on the tree that cannot be further factored. In the leftmost figure above, 27 has three leaves, all of which have the value of 3.
Now that you have completed the prime factorization, you can write the value of the ancestor, or original number, in exponential form by multiplying all of the leaves together. In the figure on the left above, the value of 27 in exponential form is 3 × 3 × 3 = 3^{3}.
Now that you understand how to move between numbers and exponents, let’s try a few questions.
EXAMPLE
What is the value of 3^{2} + 7^{3} ?
In order to answer this question, you need to find the sum of 3^{2} and 7^{3}. 3^{2} indicates that you need to multiply 3 by itself twice: 3^{2} = 3 × 3 = 9. Repeat the process for 7^{3} to find that 7^{3} = 7 × 7 × 7 = 343. Finally, add the two numbers together to find that 9 + 343 = 352.
Did you ace that question? Are you ready for a more difficult one? Here you go:
EXAMPLE
If a = 1, what is the value of a^{2} + a^{−3} ?
Unlike Example 1, this question deals with both variables and negative exponents. Never fear! The question states that a = 1, so you can substitute this value into the expression a^{2} + a^{−3} = 1^{2} + 1^{−3}. Now, take the reciprocal of 1^{−3} to transform it into positive exponent, , changing the expression 1^{2} + 1^{−3} into 1^{2} + . Since the value of 1 to any power is 1, the expression 1^{2} + = 1 + 1 = 2. Therefore, when a = 1, the value of a^{2} + a^{−3} = 2.
Another important algebra concept is converting between exponents and numbers. Let’s try a question that requires you to convert a number to an exponential expression.
EXAMPLE
Which of the following is equivalent to 156 ?
A) 2^{2} × 3^{2}
B) 2 × 3 × 13
C) 2^{2} × 3 × 13
D) 2^{2} × 3^{2} × 13
While the previous two examples focused on your ability to transform exponential expressions to numerical expressions, this question requires you to transform a number to an exponential expression. Here, you will want to create a prime factor tree where 156 is the ancestor, or original number.
Next, find two factors of 156. Since 156 is even, use 2, which goes in 78 times; the children of 156 are thus 2 and 78. Repeat this process with 78. Since 78 is even, use 2, which goes in 39 times; the children of 78 are 2 and 39.
Now repeat the process with 39. Since the digits of 39 have a sum of 12, which is a multiple to 3, 39 is a multiple of 3, which goes in 13 times; the children of 39 are therefore 3 and 13. At this point, you cannot further factor any of the numbers on your prime factor tree. So, you can now express the number 156 in exponential form by multiplying together all of the leaves, circled in the figure below.
Therefore, the expression that is equivalent to 156 is 2^{2} × 3 × 13, and the correct answer is (C).
Exponent Operations
At times, you’ll need to perform mathematical operations to simplify exponential expressions. Just like there are rules for working with numbers, there are rules for working with exponents. The specific rules you must follow when multiplying, dividing, or raising exponents to a power are as follows:
When multiplying terms that have the same base, combine the terms by adding the exponents together.
2^{3} × 2^{4} = 2^{3 + 4} = 2^{7}
6^{8} × 6^{−14} = 6^{8 + (−14)} = 6^{−6} or
a^{2} × b^{4} × a^{5} × b = a^{7}b^{5}
When dividing terms that have the same base, combine the terms by subtracting the exponents from one another.
= 4^{10−7} = 4^{3}
= 2^{5−7} = 2^{−2} or
= c^{4−2}d^{2−4} = c^{2}d^{−2} or
When an exponential term is raised to a power, you simply multiply the exponents together.
(3^{4})^{5} = 3^{4 × 5} = 3^{20}
(7^{2})^{−6} = 7^{2 × −6} = 7^{−12} or
= c^{24}d^{−6} or
One way to remember the rules for exponents is through the acronym MADSPM, which you can remember by knowing the rules of the MAD Super-Powered Monkey! Take a look at how this works:
Multiply |
x^{2} × x^{3} = x^{5} |
Divide |
= y |
Power |
(z^{2})^{3} = z^{6} |
EXAMPLE
Simplify the expression x^{4} × y^{2} × y^{3} × x^{8} .
Here you are given a string of variables that are multiplied together and asked to simplify the expression. When exponents with the same base are multiplied together, you add the exponents together. Therefore:
x^{4} × y^{2} × y^{3} × x^{8} = x^{4 + 8} × y^{2 + 3}
= x^{12} × y^{5}
= x^{12}y^{5}
That’s it—you’re done!
EXAMPLE
Which of the following is equivalent to ?
A) a^{−2}b^{−5}
B) a^{2}b^{−5}
C)
D)
To answer this question, you need to simplify the given expression. Here you are required divide exponents, so you will subtract the exponents from one another, simplifying the original expression as follows:
= a^{4 − 6} b^{3 −(−2)}
= a^{−2} b^{5}
There you go: You’ve simplified the expression! Unfortunately, a^{−2}b^{5} is not an answer choice. How can you write this expression using only positive exponents? Remember that you can convert negative exponents to positive exponents by taking the reciprocal of the negative value:
a^{−2}b^{5} =
Now that you’ve expressed the original expression using only positive exponents, you should recognize that the correct answer is (C).
Now try a question that requires you to use more than one exponent operation.
Here is how you may see exponent operations on the ACT.
Which of the following expressions is equivalent to (y^{6})^{12} ?
F. 72y
G. 12y^{6}
H. 6y^{12}
J. y^{72}
K. y^{216}
EXAMPLE
Which of the following is equivalent to ?
A) a^{8}b^{15}c^{10}
B) a^{12}b^{15}c^{30}
C)
D)
You may have noticed that there’s more going on in this question than in the previous examples; not only do you have exponents with powers, but you also have to divide exponents. According to PEMDAS, you start with the terms inside the parentheses, followed by the exponents. Since numbers with difference bases cannot directly be multiplied, there are no operations to perform within the parentheses. Therefore, the parentheses, the first operation you will perform is raising a^{2} b^{3} c^{4} to the fifth power.
Now that the parentheses have been eliminated, divide the remaining exponents. Subtract the exponents from one another, simplifying the original expression as follows:
= a^{10 − 2}b^{15}c^{20 − 10}
= a^{8}b^{15}c^{10}
Great work! You are zooming through these integer-based exponent questions. Now let’s discuss what to do in situations where you’re given an exponent that is a rational number but not an integer. For example, what if you were asked to work with the exponent x^{}? What does that even mean? Never fear, you’ll learn all about those non-integer exponents next.
Non-Integer Exponents
Up until now, we’ve focused only on integer exponents. However, exponents can be fractions, such as 1/4 or 1/2, or decimals, such as 0.25 or 0.50. In the same way we need to know certain terminology related to equations, we also need to know some vocabulary terms when working with non-integer exponents.
When working with fractional exponents, you are actually working with radicals, or expressions with roots. The symbol for radicals is . A radicand is the number that is under the radical symbol, and the index tells you the degree of the root. In the following example, k is the radicand, and n is the index.
If you have a fractional exponent, the numerator indicates the value to which the radicand should be raised, while the denominator indicates the value of the root. Take a look at the example below to see how this works out.
That’s a lot of information! And it isn’t always easy to understand examples with variables, so here are a few examples that use numbers:
Now let’s try a few questions that combine our previous knowledge of exponents with our newly acquired knowledge of fractional exponents.
This question is assessing your understanding of the rules of exponents. Whenever you raise an exponent to a power, you multiply the two exponents together. Therefore, (y^{6})^{12} = y^{6 × 12} = y^{72}. The correct answer is (J).
EXAMPLE
What is the value of 27^{} + 81^{} ?
Here you’re asked to find the value of 27^{} + 81^{}. Start by simplifying the first term:
27^{} =
= 3
Repeat the process with the second term:
81^{} =
= 9
Finally, find the sum of the individual values.
3 + 9 = 12
Therefore, the value of 27^{} + 81^{} is 12.
There is another feature of exponential terms that you should be familiar with before moving forward:
If you can express both sides of an equation as powers of the same base, you can set the exponents equal to solve for the unknown variable:
If z^{x} = z^{y}, then x = y, where z ≠ 1 and z > 0.
EXAMPLE
If c = 4 and d = 27, what is the value of c ^{} + d ^{} ?
While this question may look intimidating because of the variables involved, substituting in the values of the variables will make the question much more manageable. Since you are told that c = 4 and d = 27, you can transform the expression:
c ^{} + d ^{} = 4^{} + 27^{}
Now simplify the terms one at a time:
Repeat the process with the second term:
27^{} =
= 3
Finally, find the sum of the individual terms:
8 + 3 = 11
Therefore, the value of c ^{} + d ^{} = 8 + 3 = 11.
Let’s try one more.
Here is how you may see equations with exponents on the ACT.
What real value of x satisfies the equation 27^{x} ^{+ 2} = ?
A. −1
B.
C.
D. 2
E. 3
EXAMPLE
Simplify the expression .
You probably notice right off the bat that the question looks pretty messy. There are fractional powers to fractional powers! Don’t panic. Use PEMDAS and the rules of exponents to work through the question in steps. Start with the parentheses. Here, you told that y^{} is raised to the power of 2/3. Based on the rules of exponents, a power to a power should be multiplied. Therefore, you can get rid of the fractional exponents with the following step-by-step process:
Now that you’ve gotten rid of those funky exponents, you can divide, or subtract, the remaining exponents to simplify the expression:
= x^{2 − 1} y^{1 −(−3)}
= xy^{4}
Therefore, the expression can be simplified to xy^{4}.