Exponents

There's one more operation besides addition, subtraction, multiplication, and division. That's exponentiation, notated by superscript², or sometimes (especially on calculators and computers) by the symbol ^.

6² means "6 to the power 2", or in other words:

6² = 6 × 6 = 36. (Remember, it's not just 6 × 2.) The 6 is the base and the 2 is the exponent. This is also called "6 squared"; it's the area of a square of side 6. (See square roots.)

The exponent tells you how many times to use the base as a factor. In the example below, the base is 3 and the exponent is 4, so we find the product of four threes.

3⁴ = 3 × 3 × 3 × 3 = 9 × 9 = 81.

Example:

Which is greater, 4⁵ or 5⁴ ?

Answer:

4⁵ = 4 × 4 × 4 × 4 × 4 = 16 × 16 × 4 = 256 × 4 = 1024, while

5⁴ = 5 × 5 × 5 × 5 = 25 × 25 = 625.

To answer the question, 4⁵ is greater.

To simplify more complicated expressions, see the page on properties of exponents. This page also deals with negative and rational exponents.

Some good sequences to remember (they come up in mathematics all the time) are the square numbers and the powers of two:

Powers of 2	Square Numbers
2¹ = 2	1² = 1
2² = 4	2² = 4
2³ = 8	3² = 9
2⁴ = 16	4² = 16
2⁵ = 32	5² = 25
2⁶ = 64	6² = 36
2⁷ = 128	7² = 49
2⁸ = 256	8² = 64
2⁹ = 512	9² = 81
2¹⁰ = 1024	10² = 100

Click here for more stuff about exponent tables and patterns.

One important use of exponents is to express really large (or really small) numbers: this is called scientific notation and uses powers of ten.

Examples:

4560 = 4.56 × 10³

and

0.00003802 = 3.802 × 10^-5

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