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Chapter 2 Integers 2-10 Introducing Exponents Recall that multiplication is a short-cut method for adding a group of equal numbers. For example - 3 + 3 + 3 + 3 = 4 x 3 = 12
- 5 + 5 = 2 x 5 = 10
The same idea applies to multiplying groups of equal numbers such as 2 x 2 x 2 x 2 and 12 x 12. The short-cut method in this case uses exponential notation: - 2 x 2 x 2 x 2 = 16 expressed in exponential notation is 24 = 16
- 12 x 12 = 144 expressed in exponential notation is 122 = 144
| Definition 
| | We sometimes say a number is raised to a certain power. For example, 26 could be spoken as "two raised to the sixth power." Or we could say "two to the sixth power." Powers of 2, or "Squares" | By far, the most common exponent is 2. For instance, 32, 52, 102. Numbers that are raised to a power of two are said to be squared. Examples - "Three squared equals nine" 32 = 9
- "Five squared equals twenty five" 52 = 25
- "Ten squared equals one hundred" 102 = 100
| | | Squares of whole numbers, 0 through 25, | Exercises Cite the value of these squared integers.
Click the ? symbol to see the correct answer. | 1. 22 = ? | 2. 42 = ? | 3. 62 = ? | 4. 82 = ? | 5. 92 = ? | Powers of 3, or "Cubes" Another common exponent is 3. For instance, 33, 53, 103. Numbers that are raised to a power of three are said to be cubed. Examples - "Two cubed equals eight" 23 = 8
- "Three cubed equals twenty seven" 33 = 27
- "Ten cubed equals one thousand" 103 = 1000
Note Special Cases | Example | - 0 raised to any power equals 0
| 02 = 0 | - 1 raised to any power equals 1
| 13 = 1 | - Any value raised to the 0 power is equal to 1
| 20 = 1 | - Any value raised to the 1 power is equal to itself
| 51 = 5 | | Exponents of Signed Integers The sign of any squared value is always positive. | Example | - The square of a positive number is a positive value.
| 32 = 3 x 3 = 9 | - The square of a negative number is a positive value.
| (– 4)2 = (– 4)( – 4) = 16 |
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