# What does exponent mean

## Powers - definition and examples

### Definition of a power

A power stands for a certain arithmetic operation - it is, so to speak, a kind of mathematical abbreviation. And we can be even more precise: a power describes a multiplication.

The power describes a mathematical expression in which a number is multiplied (multiplied) by itself several times.

A power consists of a base, an exponent, which is written at the top right of the base, and the result, which is also called the power value. What is the function of these numbers? To understand this, it is worthwhile to simply write out the multiplication that describes a power.

What has happened now? If you compare the power with the written multiplication, you should notice something. As already mentioned, a power consists of a base, an exponent and a result - but which of these can be found in the written variant? In fact, we only find the base (2) and the result (8). Where did the exponent go?

The exponent as a number is not a direct part of the calculation, but only stands for the **number** the multiplication of the base by itself. Let us apply this to our example: The three does not appear as a number in the calculation - but if you count the number of twos (i.e. the base), you get the three (the exponent) again.

Everything we have learned so far can also be rewritten in a general form:

Both $ a $ and $ n $ can have negative or positive values. At the moment, however, we are initially assuming positive exponents ($ n $).

If the exponent is $ zero $, the power value is by definition $ one $, regardless of the value of the base.

The following applies: $ a ^ 0 = 1 $

### Examples of powers

(1) $ 3 ^ 3 = 3 \ times 3 \ times 3 = 27 $

(2) $ 4 ^ 4 = 4 \ times 4 \ times 4 \ times 4 = 256 $

(3) $ 4 ^ 6 = 4 \ times 4 \ times 4 \ times 4 \ times 4 \ times 4 = 4096 $

(4) $ 9 ^ 5 = 9 \ times 9 \ times 9 \ times 9 \ times 9 = 59 049 $

### Special features of exponentiation: negative numbers as a basis

Many students have problems solving potencies where the base is negative:

$ (-2) ^ 3 = (-2) \ cdot (-2) \ cdot (-2) = -8 $

$ (-2) ^ 4 = (-2) \ times (-2) \ times (-2) \ times (-2) = 16 $

At this point we have to fall back on your previous mathematical knowledge. You may remember the following rules: If you multiply two numbers with the same sign ($ + $, $ + $ or $ - $, $ - $) you get a positive number, if you multiply two numbers with different signs ($ + $, $ - $ or $ - $, $ + $) you get a negative number.

Two simple rules can therefore be derived for powers:

- If the base is a negative number and the exponent is odd, the result is a negative number.

- If the base is a negative number and the exponent is even, the result is a positive number.

- If the base is a negative number and the exponent is odd, the result is a negative number.
- If the base is a negative number and the exponent is even, the result is a positive number.

*Deepen your newly learned knowledge in the exercises. Good luck with it!*

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