Walking through a potential apartment, house or office space, you use exponents, perhaps subconsciously, whenever you consider total and component area sizes. One square foot, for example, is 1 foot by 1 foot, or 1 foot raised to the exponent (or power of) 2, or (1 foot)^2. You also use exponents when specifying computer RAM in gigabytes (10^9 bytes) or external hard drive storage capacity in terabytes (10^12 bytes).

## Definitions

### Step 1

Recognize an exponent and its base. An exponent indicates the number of times to multiply a number (or base) by itself.

Examples: The number 4 raised to the power of (or exponent) 3 = 4^3 = 4 x 4 x 4 = 64. 12^5 = 12 x 12 x 12 x 12 x 12 = 248,832.

### Step 2

Learn the basic rule for the exponent 0 (zero). Any number (with the possible exception of 0) raised to the exponent (or power) 0 equals 1.

Examples: 1^0 = 1.

10^0 = 1.

673^0 = 1.

### Step 3

Learn the meaning of a negative exponent. The top number in a simple fraction displayed in stacked format (or, in single-line format, the expression preceding the slash, “/”) is the numerator. The bottom number is the denominator. For example, in the fraction 1/2, the numerator is 1 and the denominator is 2. When a whole number is raised to a power containing a negative number, it becomes a fraction. The value of the numerator equals 1, and value of the denominator equals the whole number raised to the power without the minus sign. When a fraction is raised to a power containing a negative number, the expressions in the numerator and denominator switch positions.

Examples: 5^-3 = 1/5^3 = 1/(5 x 5 x 5) = 1/125.

(1/4)^-3 = 4^3 = 4 x 4 x 4 = 64.

(4/5)^-1 = 5/4 = 1.25.

(4/5)^-3 = (5 x 5 x 5)/(4 x 4 x 4) = 125/64 = 1.95 (with rounding to 3 digits).

## Basic Rules, Shortcuts and Exercises

### Step 1

Multiply by a number with the exponent 0. Any number with the exponent 0 equals 1.

Example: 4^0 x 4^3 = 1 x (4 x 4 x 4) = 1 x 64 = 64.

### Step 2

Multiply by another number.

Example: 10^3 x 4.3 = (10 x10 x 10) x 4.3 = 1,000 x 4.3 = 4,300.

### Step 3

Multiply a number raised to a negative power by another number.

Example: 10^-3 x 4.3 = 1/(10 x 10 x 10) x 4.3 = 1/1000 x 4.3 = 0.0043.

### Step 4

Multiply two numbers with the same base and different exponents.

A shortcut is to add the exponents.

Example: 4^3 x 4^6 = 4^(3 + 6) = 4^9 = 262,144.

### Step 5

Calculate the value of a number with an exponent raised to a power.

A shortcut is to multiply the exponents.

Example: (3^2)^4 = 3^(2 X 4) = 3^8 = 6,561.

### Step 6

Multiply two different numbers raised to different powers.

Example: 3^4 x 6^3 = (3 x 3 x 3 x 3) x (6 x 6 x 6) = 81 x 216 = 17,496.