Lesson Objectives
- Demonstrate an understanding of multiplication with whole numbers
- Demonstrate an understanding of multiplication by powers of 10 (trailing zeros)
- Learn how to write the repeated multiplication of the same whole number using exponents
- Learn how to evaluate an exponential expression
- Learn how to quickly evaluate 10 raised to a whole number exponent
What are Exponents?
Once we have a good understanding of operations with whole numbers (addition, subtraction, multiplication, and division), we are ready to move on to exponents. Whole number exponents larger than 1 are used to conveniently notate repeated multiplication of the same number. Suppose we had the following scenario:
2 x 2 x 2 x 2 x 2
This can be written more compactly using an exponent as:
25
The 2 is referred to as the base. This is the larger number and represents the number that is multiplied by itself in the repeated multiplication. The 5 is referred to as the exponent or sometimes as the power. This is the smaller number placed at the top right of the base. A whole number exponent larger than 1 tells us how many factors of the base are present in the repeated multiplication.
Let's take a look at a few examples:
Example 1: Write 3 x 3 x 3 x 3 x 3 x 3 in exponential form.
3 - will be our base, this is the number being multiplied by itself in the repeated multiplication
6 - will be our exponent, this is the number of factors of the base in the repeated multiplication
36
Example 2: Write 11 x 11 x 11 in exponential form.
11 - will be our base, this is the number being multiplied by itself in the repeated multiplication
3 - will be our exponent, this is the number of factors of the base in the repeated multiplication
113
171 = 17
9751 = 975
An exponent of 0, on a base that is not 0, results in 1. The math behind this is a bit advanced for this stage, but we will cover this property in algebra 1 under exponents. For now, if you see any nonzero number raised to the power of 0, the result is always 1:
2,355,7770 = 1
3990 = 1
4 x 4 x 4
Now we can multiply:
4 x 4 x 4 = 64.
Example 3: Evaluate 34 and 192.
Write 34 as a repeated multiplication and multiply:
3 x 3 x 3 x 3 = 81
34 = 81
Write 192 as a repeated multiplication and multiply:
19 x 19 = 361
192 = 361
Cutoff the trailing zeros and multiply the remaining numbers
Attach the total number of trailing zeros between all factors to the answer
102 = 10 x 10 = 100
1 x 1 = 1, then attach two zeros to the end. This gives us 100
103 = 10 x 10 x 10 = 1000
1 x 1 x 1 = 1, then attach three zeros to the end. This gives us 1000
In each case where 10 is the base, cutting off the trailing zeros gives us a multiplication problem where 1 is the only factor. It does not matter how many factors of 1 are present, the result of the multiplication will always be 1. Therefore, we only need to know how many trailing zeros will be involved. For this, we can simply look at the exponent. We saw that 102 = 10 x 10, which has two factors of 10 and a total of 2 trailing zeros, and 103 = 10 x 10 x 10, which has three factors of 10 and a total of 3 trailing zeros. Therefore, we can form a shortcut. To quickly evaluate any exponential expression where 10 is raised to a whole number larger than 1:
Start by writing the number 1
Attach trailing zeros to the end of 1 equal to the exponent. Let's try an example:
Example 4: Evaluate 104.
Write the number 1:
1
Attach trailing zeros to the end of 1 equal to the exponent, which is 4 in this case:
10,000
104 = 10,000
2 to the 5th power
Generally, we say the base followed by to the __ power, where the blank is the exponent. Not every exponential expression will sound the same out loud. In the case of an exponent of 2 or 3, we have special names:
A number raised to the second power is said to be "squared"
A number raised to the third power is said to be "cubed"
22 can be read as 2 to the 2nd power or 2 squared
23 can be read as 2 to the 3rd power or 2 cubed
Example 5: How would we read 49 and 623?
49 - Our base is 4, and our exponent is 9. We would say:
4 to the 9th power
623 - Our base is 6, and our exponent is 23. We would say:
6 to the 23rd power
2 x 2 x 2 x 2 x 2
This can be written more compactly using an exponent as:
25
The 2 is referred to as the base. This is the larger number and represents the number that is multiplied by itself in the repeated multiplication. The 5 is referred to as the exponent or sometimes as the power. This is the smaller number placed at the top right of the base. A whole number exponent larger than 1 tells us how many factors of the base are present in the repeated multiplication.
Let's take a look at a few examples: Example 1: Write 3 x 3 x 3 x 3 x 3 x 3 in exponential form.
3 - will be our base, this is the number being multiplied by itself in the repeated multiplication
6 - will be our exponent, this is the number of factors of the base in the repeated multiplication
36
Example 2: Write 11 x 11 x 11 in exponential form.
11 - will be our base, this is the number being multiplied by itself in the repeated multiplication
3 - will be our exponent, this is the number of factors of the base in the repeated multiplication
113
Exponent of 1 or 0
When working with whole-number exponents, we have two special case scenarios: An exponent of 1, just gives us the base as an answer:171 = 17
9751 = 975
An exponent of 0, on a base that is not 0, results in 1. The math behind this is a bit advanced for this stage, but we will cover this property in algebra 1 under exponents. For now, if you see any nonzero number raised to the power of 0, the result is always 1:
2,355,7770 = 1
3990 = 1
Evaluating Exponential Expressions
In some cases, we need to evaluate or find the value of an exponential expression. In order to do this, we write an exponential expression as a repeated multiplication and then perform the multiplication. How could we find the value of 43? First, write as a repeated multiplication. We see that we have a base of 4 and an exponent of 3. This means we will have 3 factors of 4 or:4 x 4 x 4
Now we can multiply:
4 x 4 x 4 = 64.
Example 3: Evaluate 34 and 192.
Write 34 as a repeated multiplication and multiply:
3 x 3 x 3 x 3 = 81
34 = 81
Write 192 as a repeated multiplication and multiply:
19 x 19 = 361
192 = 361
Exponents with a base of 10
When we have 10 as the base and a whole number exponent larger than 1, we can use a shortcut. We previously learned how to multiply by powers of 10 (numbers with trailing zeros):Cutoff the trailing zeros and multiply the remaining numbers
Attach the total number of trailing zeros between all factors to the answer
102 = 10 x 10 = 100
1 x 1 = 1, then attach two zeros to the end. This gives us 100
103 = 10 x 10 x 10 = 1000
1 x 1 x 1 = 1, then attach three zeros to the end. This gives us 1000
In each case where 10 is the base, cutting off the trailing zeros gives us a multiplication problem where 1 is the only factor. It does not matter how many factors of 1 are present, the result of the multiplication will always be 1. Therefore, we only need to know how many trailing zeros will be involved. For this, we can simply look at the exponent. We saw that 102 = 10 x 10, which has two factors of 10 and a total of 2 trailing zeros, and 103 = 10 x 10 x 10, which has three factors of 10 and a total of 3 trailing zeros. Therefore, we can form a shortcut. To quickly evaluate any exponential expression where 10 is raised to a whole number larger than 1:
Start by writing the number 1
Attach trailing zeros to the end of 1 equal to the exponent. Let's try an example:
Example 4: Evaluate 104.
Write the number 1:
1
Attach trailing zeros to the end of 1 equal to the exponent, which is 4 in this case:
10,000
104 = 10,000
Reading Exponents
How do we read exponential expressions aloud? If we see something such as 25, we can read this as:2 to the 5th power
Generally, we say the base followed by to the __ power, where the blank is the exponent. Not every exponential expression will sound the same out loud. In the case of an exponent of 2 or 3, we have special names:
A number raised to the second power is said to be "squared"
A number raised to the third power is said to be "cubed"
22 can be read as 2 to the 2nd power or 2 squared
23 can be read as 2 to the 3rd power or 2 cubed
Example 5: How would we read 49 and 623?
49 - Our base is 4, and our exponent is 9. We would say:
4 to the 9th power
623 - Our base is 6, and our exponent is 23. We would say:
6 to the 23rd power
Skills Check:
Example #1
Evaluate each. $$5^{4}$$
Please choose the best answer.
A
625
B
125
C
25
D
100
E
225
Example #2
Evaluate each. $$10^{5}$$
Please choose the best answer.
A
10
B
100,000
C
1,000
D
1,000,000
E
100
Example #3
Write in exponential form. $$4 \cdot 4 \cdot 4 \cdot 4 \cdot 4$$
Please choose the best answer.
A
$$5^{4}$$
B
$$4 + 5^{4}$$
C
$$4^{3}$$
D
$$4^{5}$$
E
$$4 + 5^{1}$$
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