The square of a number is that number times itself. 5 squared, denoted 5^{2} , is equal to 5×5 , or 25. 2 squared is 2^{2} = 2×2 = 4 . One way to remember the term "square" is that there are two dimensions in a square (height and width) and the number being squared appears twice in the calculation. In fact, the term "square" is no coincidence--the square of a number is the area of the square with sides equal to that number.
A number that is the square of a whole number is called a perfect square. 4^{2} = 16 , so 16 is a perfect square. 25 and 4 are also perfect squares. We can list the perfect squares in order, starting with 1^{2} : 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, ...
The cube of a number is that number times itself times itself. 5 cubed, denoted 5^{3} , is equal to 5×5×5 , or 125. 2 cubed is 2^{3} = 2×2×2 = 8 . The term "cube" can be remembered because there are three dimensions in a cube (height, width, and depth) and the number being cubed appears three times in the calculation. Similar to the square, the cube of a number is the volume of the cube with sides equal to that number--this will come in handy in higher levels of math.
The "2" in " 5^{2} " and the "3" in " 5^{3} " are called exponents. An exponent indicates the number of times we must multiply the base number. To compute 5^{2} , we multiply 5 two times (5×5) , and to compute 5^{3} , we multiply 5 three times (5×5×5) .
Exponents can be greater than 2 or 3. In fact, an exponent can be any number. We write an expression such as " 7^{4} " and say "seven to the fourth power." Similarly, 5^{9} is "five to the ninth power," and 11^{56} is "eleven to the fifty-sixth power."
Since any number times zero is zero, zero to any (positive) power is always zero. For example, 0^{31} = 0 .
A number to the first power is that number one time, or simply that number: for example, 6^{1} = 6 and 53^{1} = 53 . We define a number to the zero power as 1: 8^{0} = 1 , (- 17)^{0} = 1 , and 521^{0} = 1 .
Here is a list of the powers of two:
2^{0} | = | 1 | |
2^{1} | = | 2 | |
2^{2} | = | 2×2 = 4 | |
2^{3} | = | 2×2×2 = 8 | |
2^{4} | = | 2×2×2×2 = 16 | |
2^{5} | = | 2×2×2×2×2 = 32 |
Here is a list of the powers of ten:
10^{0} | = | 1 | |
10^{1} | = | 10 | |
10^{2} | = | 10×10 = 100 | |
10^{3} | = | 10×10×10 = 1, 000 | |
10^{4} | = | 10×10×10×10 = 10, 000 | |
10^{5} | = | 10×10×10×10×10 = 100, 000 |
Look familiar? 10^{0} is 1 one (a 1 in the ones place), 10^{1} is 1 ten (a 1 in the tens place), 10^{2} is 1 hundred, 10^{3} is 1 thousand, 10^{4} is 1 ten thousand, etc. This is the meaning of base ten--a "1" in each place represents a number in which the base is 10 and the exponent is the number of zeros after the 1. The place value is the number that is multiplied by this number. For example, a 5 in the thousands place is equivalent to 5×1000 , or 5×10^{3} .
We can write out any number as a sum of single-digit numbers times powers of ten. The number 492 has a 4 in the hundreds place (4×10^{2}) , a 9 in the tens place (9×10^{1}) and a 2 in the ones place (2×10^{0}) . Thus, 492 = 4×10^{2} +9×10^{1} +2×10^{0} .
Examples: Write out the following numbers as single-digit numbers times
powers of ten.
935 = 9×10^{2} +3×10^{1} +5×10^{0}
67, 128 = 6×10^{4} +7×10^{3} +1×10^{2} +2×10^{1} +8×10^{0}
4, 040 = 4×10^{3} +0×10^{2} +4×10^{1} +0×10^{0}