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Powers, Exponents, and Roots

Squares, Cubes, and Higher Order Exponents

Terms

Squares, Cubes, and Higher Order Exponents, page 2

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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 .

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