Roots and Exponents

Fundamentals of Social Statistics by Adam J. McKee

In statistics, there is a lot of squaring.  To square a number, simply multiply the number by itself.

Example: 32 = 3 x 3 = 9

Note that we know to multiply 3 by itself in the above example because the raised-up number 2—the exponent—tells us to multiply the number preceding the exponent by itself that number of times. This can also be expressed as “three to the second power.”

We can raise a number to the third power, called cubing.

Example:  23 = 2 x 2 x 2 = 8

Special Exponent Rules

-Any number raised to the first power equals itself.

-If any exponent appears outside of parentheses, then any operations inside the parentheses are done first.

-If a negative number is raised to an exponent, the result will be positive for exponents that are even and negative for exponents that are odd.

-An exponent applies only to the base that is just in front of it.

A number printed above the line like an exponent is called a superscript.  This is the term to look for when formatting an exponent on your computer.

Taking a square root is the opposite of squaring.  This means, for example, that the square root of 9 is 3.  The formal name for the square root sign (√) is the radical sign.

When a radical sign appears in an equation (as it often does in statistics), it has the same effect as parentheses on the order of operations.


Operations Using Computer Notation

Computer notation offers a concise way to depict mathematical operations. Let’s delve into how you can express squaring, cubing, taking the square root, and the cube root of a number using this notation.

Squaring a Number: Squaring refers to multiplying a number by itself. In computer notation, it’s often expressed using the “^” (caret) symbol. For instance, to square the number 5, you would write 5^2, which is equal to 25.

Cubing a Number: Similarly, cubing means multiplying a number by itself twice. Using the caret symbol again, the cube of 5 would be written as 5^3, resulting in 125.

Square Root: The square root of a number finds a value that, when multiplied by itself, gives the original number. In computer notation, the square root is often symbolized by “sqrt.” So, to express the square root of 25, you’d write sqrt(25), which equals 5.

Cube Root: The cube root is a value that, when multiplied by itself two more times, returns the original number. Computer notation often uses “cbrt” to denote cube root. Thus, the cube root of 27 would be represented as cbrt(27), which is 3.

To make these calculations on platforms like Excel:

  1. For Squaring and Cubing: Use the caret (^) symbol. For instance, A1^2 will square the value in cell A1, and A1^3 will cube it.
  2. For Square and Cube Roots: Use the SQRT() and POWER() functions respectively. For the square root of a value in cell A1, you’d write =SQRT(A1). For its cube root, use =POWER(A1,1/3).

By understanding this computer notation and these Excel functions, you can efficiently perform these advanced mathematical operations on any platform.

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Last Modified:  08/18/2023

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