Exponents Let s start with a review of the basics. 2 5 = 2 2 2 2 2 When writing 2 5, the 2 is the base, and the 5 is the exponent or power. We generally think of multiplication when we see a number with an exponent. With 2 5, we think of five twos multiplied together. So let s make a list to practice this way of thinking. 2 2 = 2 2 = 4 2 3 = 2 2 2 = 8 2 4 = 2 2 2 2 = 6 2 5 = 2 2 2 2 2 = 32 2 6 = 2 2 2 2 2 2 = 64 2 7 = 2 2 2 2 2 2 2 = 28 2 8 = 2 2 2 2 2 2 2 2 = 256 2 9 = 2 2 2 2 2 2 2 2 2 = 52 2 0 = 2 2 2 2 2 2 2 2 2 2 = 024 2 = 2 2 2 2 2 2 2 2 2 2 2 = 2048 2 2 = 2 2 2 2 2 2 2 2 2 2 2 2 = 4096 2 3 = 2 2 2 2 2 2 2 2 2 2 2 2 2 = 892 2 4 = 2 2 2 2 2 2 2 2 2 2 2 2 2 2 = 6,384 So, obviously, exponents are all about multiplication. Or, are they?
Let s reverse the order of our list. 2 4 = 2 2 2 2 2 2 2 2 2 2 2 2 2 2 = 6,384 2 3 = 2 2 2 2 2 2 2 2 2 2 2 2 2 = 892 2 2 = 2 2 2 2 2 2 2 2 2 2 2 2 = 4096 2 = 2 2 2 2 2 2 2 2 2 2 2 = 2048 2 0 = 2 2 2 2 2 2 2 2 2 2 = 024 2 9 = 2 2 2 2 2 2 2 2 2 = 52 2 8 = 2 2 2 2 2 2 2 2 = 256 2 7 = 2 2 2 2 2 2 2 = 28 2 6 = 2 2 2 2 2 2 = 64 2 5 = 2 2 2 2 2 = 32 2 4 = 2 2 2 2 = 6 2 3 = 2 2 2 = 8 2 2 = 2 2 = 4 Now, take a look at the products from the top of the list to the bottom. Instead of multiplying by two, we are dividing by 2 to get the next one. So, let s continue the list by dividing by 2. 2 = 2 2 0 = 2 = 2 2 2 = 4
Any multiplication pattern, in reverse, becomes a division pattern. When applying the division pattern to exponents, it proves a few things.. It shows why a base to the power of is itself. Example: 5 2 = 25 25 5 = 5 Therefore, 5 = 5. You can also think of 5 as one 5, but that doesn t help you extend to the power of zero or negative exponents. 2. It also shows why bases to the power of 0 equal. Example: 5 2 = 25 25 5 = 5 5 = 5 5 5 = Therefore, 5 0 =. *There is an exception. If the base is 0, and since you have to use a division pattern to get to a base with the power of 0, then you have to divide zero by zero. What do we get when we divide any number by 0? It s undefined. You cannot divide by zero. So, 0 0 = undefined. Not all mathematicians agree with this, but for middle school, this is what we can understand and support with division. 3. Any base to the power of a negative exponent is the reciprocal of the same base to the opposite, positive exponent. Example: 5 2 = 25 and 52 = 25 Example: 2 3 = 8 and 23 = 8
Let s do some examples using rational numbers with exponents. Example: 5 5 5 = 5 3 = 25 The answer, 5 3, is how you would write the expression using exponents. The answer, 25, is what you write if you are asked to evaluate the expression. Example: ( 2 3 )3 = ( 2 ) 3 (2) 3 (2) = 2 2 2 = 8 3 3 3 3 27 Now, let s use variables with exponents. Example: 3 3 3 d d d d = 3 3 d 4 = 27d 4 Example: g g h h h = g 2 h 3 Another aspect of exponents that confuses a lot of people involves negatives. For example, ( 4) ( 4) = ( 4) 2 = 6, but 4 2 = 6. There is a big difference between ( 4) 2 and 4 2. Read ( 4) 2 as negative four, to the power of two. Read 4 2 as the negative of, four to the power of two. Make sure you pause for the comma when saying it. ( 4) 2 in expanded form is ( 4) ( 4). 4 2 in expanded form is (4 4). Example: 3 4 = (3 3 3 3) = (8) = 8 Example: ( 3) 4 = ( 3) ( 3) ( 3) ( 3) = 8 Example: ( 3 7 )3 = ( 3 7 ) ( 3 7 ) ( 3 7 ) = ( 3) ( 3) ( 3) 7 7 7 Example: ( 3 5 )2 = ( 3 5 3 5 ) = (3 3) (5 5) = 9 25 = 27 343 Example: ( 4 )2 = ( ) ( ) = 4 4 ( ) 4 ( ) = ( ) ( ) = 4 4 4 6 Example: g g ( h) ( h) ( h) = g 2 ( h) 3
You can use exponents to do prime factorization of composite numbers. 400 2 200 2 2 00 2 2 2 50 2 2 2 2 25 2 2 2 2 5 5 So, 400 = 2 4 5 2 850 2 425 2 5 85 2 5 5 7 So, 850 = 2 5 2 7 Notice, in both answers, all of the bases are prime numbers and they are listed least to greatest. 800 8 00 9 9 0 0 3 3 3 3 2 5 2 5 So, 800 = 2 2 3 4 5 2
You can even use exponents to solve word problems where something is increasing or decreasing at a constant rate. Let s pretend we are designing stacking rings like a baby/toddler might have. The base ring has a diameter of 6 centimeters. The next ring s diameter will be 7 8 of the diameter of the one below it. If our stack contains five rings in all, what will be the diameter of the top ring. So, the one above the bottom ring will have a diameter of 6cm 7 8. The one above it, our middle ring, will have a diameter of 6cm 7 8 7 8. The one above it, our ring that is next to the top one, has a diameter of 6cm 7 8 7 8 7 8. The top ring will have a diameter of 6cm 7 8 7 8 7 8 7 8. How many rings did we have? 5 How many factors do we have? 5 Each ring is represented in our expression. So, if our answer is supposed to be in terms of exponents, the top ring has a diameter of 6cm ( 7 8 )4. If our item is increasing in size, your factor/base that is raised to a power, will be greater than one. If it decreases, the base is less than one.
Let s not forget about using order of operations to evaluate expressions containing exponents. Example: 6 4 2 3 6 4 2 3 Exponents must be evaluated first. 2 3 = 8 6 4 8 Then, multiplication. 4 8 = 32 6 32 Finally, subtraction. 6 32 = 26 26 2 Example: 2 3 (33 5 2 4 ) 3 (33 3 2 4 ) Exponents must be evaluated first, left to right. 3 3 = 27 2 3 (27 3 24 ) 2 4 = 6 2 3 2 3 2 3 (27 3 6) Then, multiplication inside the parentheses. 3 6 = 48 (27 48) Then, subtraction inside the parentheses. 27 48 = 2 ( 2) Rewrite the integer as an improper fraction. 2 2 3 42 3 Multiply. Simplify. 4
There are some properties of exponents you must practice to commit to memory. Product of Powers Property To multiply powers with the same base, you add the exponents. Example: 5 5 5 2 5 5 5 2 5 5+2 5 7 This makes sense, since 5 5 5 2 = (5 5 5 5 5) (5 5) = 5 5 5 5 5 5 5 = 5 7 Example: (4 2 ) 3 (4 2 ) 3 4 2 4 2 4 2 4 2+2+2 4 6 Example: ( 3mn) 4 ( 3mn) 4 ( 3mn) ( 3mn) ( 3mn) ( 3) m n ( 3) m n ( 3) m n ( 3) m n ( 3)( 3)( 3)( 3) m m m m n n n n ( 3) 4 m 4 n 4 8m 4 n 4
Example: ( 6 k)2 ( 6 k)2 ( 6 k) ( 6 k) ( 6 ) k ( 6 ) k ( ) ( ) k k 6 6 k k 6 6 ( ) ( ) 6 6 36 k k k k 36 k2 Example: James has 5 2 marbles. His brother has five times that many. In terms of exponents, how many does his brother have? 5 2 5 5 2 5 5 2+ 5 3 His brother has 5 3 marbles.
Quotient of Powers Property To divide powers with the same base, you subtract the exponents. Example: Simplify the expression and write your answer as a power. 5 5 5 2 or 55 5 5 5 5 5 2 5 5 2 5 3 This makes sense, since = 5 5 5 5 5 5 2 5 5 = 5 5 5 = 53 = 53 Example: Simplify the expression. Write your answer as a power. p 5 p 8 or p5 p 5 p 8 p 5 8 p 7 Example: Simplify the expression. Write your answer as a power. 93 9 2 9 3 9 2 9 5 9 6 9 3+2 9 5+6 9 5 9 9 5 9 4 9 5 9 6 5 2 p 8
Example: Simplify the expression. 27 h 5 2 7 h 5 2 4 2 7 4 h 5 2 3 h 5 8 h 5 8h 5 Example: Simplify the expression. 36 m 8 n 5 3 6 m 8 m 4 n 5 3 4 m 2 n 3 3 6 m 8+4 n 5 3 4 m 2 n 3 3 6 m 2 n 5 3 4 m 2 n 3 2 4 3 4 m 2 n 3 3 6 4 m 2 2 n 5 3 3 2 m 0 n 2 9m 0 n 2 Example: Simplify the expression. j5 k 2 l 9 j 5 k 2 l 9 j 2 l 5 j 5 2 k 2 l 9 5 j 3 k 2 l 4 j 2 l 5
Negative and Zero Exponents The chart below shows positive integer bases, ranging from two to ten, with exponents ranging from positive three to negative three. 2³ = 8 3³ = 27 4³ = 64 5³ = 25 6³ = 26 7³ = 343 8³ = 52 9³ = 729 0³ = 000 2² = 4 3² = 9 4² = 6 5² = 25 6² = 36 7² = 49 8² = 64 9² = 8 0² = 00 2¹ = 2 3¹ = 3 4¹ = 4 5¹ = 5 6¹ = 6 7¹ = 7 8¹ = 8 9¹ = 9 0¹ = 0 2⁰ = 3⁰ = 4⁰ = 5⁰ = 6⁰ = 7⁰ = 8⁰ = 9⁰ = 0⁰ = 2 ¹ = 2 2 ² = 4 2 ³ = 8 3 ¹ = 3 4 ¹ = 4 3 ² = 9 4 ² = 6 3 ³ = 27 4 ³ = 64 5 ¹ = 5 5 ² = 25 5 ³ = 25 6 ¹ = 6 6 ² = 36 6 ³ = 26 7 ¹ = 7 7 ² = 49 7 ³ = 343 8 ¹ = 8 8 ² = 64 8 ³ = 52 9 ¹ = 9 9 ² = 8 9 ³ = 729 0 ¹ = 0 0 ² = 0 ³ = 00 000 Let s examine the chart and look for patterns. As mentioned previously, non-zero numbers to the power of zero equal. This is shown in the row highlighted green. Notice how the purple row and the red row are reciprocals. The dark blue and orange rows are reciprocals. The light blue row and the yellow row are reciprocals. That means that any quantity to a negative power can be written as its reciprocal to the opposite, positive exponent. *Final answers should only contain positive exponents. Example: 5 4 = 5 4 = 625 Example: ( 5) 7 ( 5) 7 = ( 5) 7+( 7) = ( 5) 0 = Example: 6 3 6 5 6 9 6 6 3+5 62 6 = 9+( 6) = 6 6 3 = 62 3 = 6 = 6 = 6 Example: a 9 = a 9 Example: 8b 5 b 5 = 8b 5 5 = 8b 0 = 8 b 0 Example: a 5 b 2 = a5 = a5 b 2 b 2
Example: a 2 b 3 = a2 b 3 = a2 b 3 = a2 = a2 b 3 b 3 Example: a 4 b 3 c 5 d 6 = a2 b 3 c 5 d 6 = a2 b 3 a 2 d 6 b 3 c 5 c 5 d6 = a2 b 3 d6 = c 5 ³ = 0³ = 0 ² = 0² = 0 ¹ = 0¹ = 0 ⁰ = 0⁰ = undefined ¹ = = ¹ 0 ¹ = undefined ² = = ² 0 ² = undefined ³ = = ³ 0 ³ = undefined Notice that to any power(exponent) equals. Zero to the power of any natural number (positive whole numbers) is 0. Zero to the power of zero or a negative power(exponent) is undefined, because to get to the zero power or a negative exponent, you have to divide by zero, Any number divided by zero is undefined.
So, if you want to extend this concept a bit, look at the next four charts. ( ) ³ = ³ 2 2³ 8 2³ = 8 ( ) ² = ² 2 2² 4 2² = 4 ( ) ¹ = ¹ 2 2¹ 2 2¹ = 2 ( ) ⁰ = ⁰ 2 2⁰ 2⁰ = ( ¹ ) ¹ = = 2¹ = 2 = 2 2 ¹ = = 2 2 ¹ ¹ 2 2¹ ( ² ) ² = = 2² = 4 = 4 2 ² = = 2 2 ² ² 4 2² ( ³ ) ³ = = 2³ = 8 = 8 2 ³ = = 2 2 ³ ³ 8 2³ ( ) ³ = ³ 3 3³ 27 3³ = 27 ( ) ² = ² 3 3² 9 3² = 9 ( ) ¹ = ¹ 3 3¹ 3 3¹ = 3 ( ) ⁰ = ⁰ 3 3⁰ 3⁰ = ( ¹ ) ¹ = = 3¹ = 3 = 3 3 ¹ = = 3 3 ¹ ¹ 3 3¹ ( ² ) ² = = 3² = 9 = 9 3 ² = = 3 3 ² ² 9 3² ( ³ ) ³ = = 3³ = 27 = 27 3 ³ = = 3 3 ³ ³ 27 3³ ( ) ³ = ³ 4 4³ 64 4³ = 64 ( ) ² = ² 4 4² 6 4² = 6 ( ) ¹ = ¹ 4 4¹ 4 4¹ = 4 ( ) ⁰ = ⁰ 4 4⁰ 4⁰ = ( ¹ ) ¹ = = 4¹ = 4 = 4 4 ¹ = = 4 4 ¹ ¹ 4 4¹ ( ² ) ² = = 4² = 6 = 6 4 ² = = 4 4 ² ² 6 4² ( ³ ) ³ = = 4³ = 64 = 64 4 ³ = = 4 4 ³ ³ 64 4³
( ) ³ = ³ 5 5³ 25 5³ = 25 ( ) ² = ² 5 5² 25 5² = 25 ( ) ¹ = ¹ 5 5¹ 5 5¹ = 5 ( ) ⁰ = ⁰ 5 5⁰ 5⁰ = ( ¹ ) ¹ = = 5¹ = 5 = 5 5 ¹ = = 5 5 ¹ ¹ 5 5¹ ( ² ) ² = = 5² = 25 = 25 5 ² = = 5 5 ² ² 25 5² ( ³ ) ³ = = 5³ = 25 = 25 5 ³ = = 5 5 ³ ³ 25 5³ When you have a fraction to a power, it can be rewritten as the numerator to the power over the denominator to the power. Any base number to a power can be rewritten as the reciprocal of the base to the opposite power.