Powers, Exponents, and Roots

Squares, Cubes, and Higher Order Exponents

Squares

The square of a number is that number times itself. 5 squared, denoted 52 , is equal to 5×5 , or 25. 2 squared is 22 = 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. 42 = 16 , so 16 is a perfect square. 25 and 4 are also perfect squares. We can list the perfect squares in order, starting with 12 : 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, ...

Cubes

The cube of a number is that number times itself times itself. 5 cubed, denoted 53 , is equal to 5×5×5 , or 125. 2 cubed is 23 = 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.

Exponents

The "2" in " 52 " and the "3" in " 53 " are called exponents. An exponent indicates the number of times we must multiply the base number. To compute 52 , we multiply 5 two times (5×5) , and to compute 53 , 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 " 74 " and say "seven to the fourth power." Similarly, 59 is "five to the ninth power," and 1156 is "eleven to the fifty-sixth power."

Since any number times zero is zero, zero to any (positive) power is always zero. For example, 031 = 0 .

A number to the first power is that number one time, or simply that number: for example, 61 = 6 and 531 = 53 . We define a number to the zero power as 1: 80 = 1 , (- 17)0 = 1 , and 5210 = 1 .

Here is a list of the powers of two:

 20 = 1 21 = 2 22 = 2×2 = 4 23 = 2×2×2 = 8 24 = 2×2×2×2 = 16 25 = 2×2×2×2×2 = 32

and so on...

Exponents and the Base Ten System

Here is a list of the powers of ten:

 100 = 1 101 = 10 102 = 10×10 = 100 103 = 10×10×10 = 1, 000 104 = 10×10×10×10 = 10, 000 105 = 10×10×10×10×10 = 100, 000

and so on...

Look familiar? 100 is 1 one (a 1 in the ones place), 101 is 1 ten (a 1 in the tens place), 102 is 1 hundred, 103 is 1 thousand, 104 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 5×103 .

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 (4×102) , a 9 in the tens place (9×101) and a 2 in the ones place (2×100) . Thus, 492 = 4×102 +9×101 +2×100 .

Examples: Write out the following numbers as single-digit numbers times powers of ten.

935 = 9×102 +3×101 +5×100
67, 128 = 6×104 +7×103 +1×102 +2×101 +8×100
4, 040 = 4×103 +0×102 +4×101 +0×100

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