# What is 8 8 4 2

## Powers / exponentiates numbers

The exponentiation in mathematics is an abbreviated notation. How exactly this can be used to abbreviate a multiplication is explained below using a few examples. In addition, we offer exercises and old written exams with solutions.

In order to abbreviate multiplications in mathematics and thus represent very large and very small numbers, the so-called powers were introduced. The easiest way to explain this is through examples:

• 102 = 10 · 10
• 103 = 10 · 10 · 10
• 104 = 10 · 10 · 10 · 10

As you can see, it says 102 the number 10 twice there with a mark in between. At 103 the number 10 is written down and multiplied three times, etc.

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### Base and exponent

Time to sort out some general issues. The representation is started with an carried out. In our example above, the a was the number 10. The lowercase n above was 2 or 3 or 4 in our example. The a is usually referred to as the base, the n is the exponent (also called the exponent). Here are a few more examples to clarify:

• 23 = 2 · 2 · 2 = 8
• 74 = 7 · 7 · 7 · 7 = 2401
• 35 = 3 · 3 · 3 · 3 · 3 = 243
• 84= 8 · 8 · 8 · 8 = 4096

The examples also show quite well: Even larger numbers can be represented very nicely and easily using the power notation.

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### Powers: Negative numbers + laws of calculation

Knowledge of variables is necessary to understand the following section. So I recommend reading our article on variables. But first we turn to negative numbers and floating point / decimal numbers. The following examples show this:

• 1,24 = 1,2 · 1,2 · 1,2 · 1,2 = 2,0736
• 2,343 = 2,34 · 2,34 · 2,34 = 12,812904
• (-3)4 = (-3) · (-3) · (-3) · (-3) = 81
• (-1,4)2= (-1,4) · (-1,4) = 1,96

Laws of calculation:

The following arithmetic laws apply to multiplying powers: Powers with the same base are multiplied by adding the exponents and keeping the base. The following is the general calculation rule and two examples:

an · Am = an + m

Examples:

• 25 · 23 = 25+3 = 28 = 256
• 4-3 · 47 = 4-3+7 = 44 = 256

However, this law must not be confused with another law of power calculation. This means: Powers with the same exponent are multiplied by multiplying the bases and keeping the exponent. The following is the general calculation rule and two examples:

an · Bn = (a b)n

Examples:

• 53 · 23 = (5 · 2)3 = 103 = 1000
• 35 · 25 = (3 · 2)5 = 65 = 7776

And we do not want to withhold another law of power from you in this context. This says: A power is raised to the power by multiplying the exponents and keeping the base. The following is the general calculation rule and an example:

(an)m = an · m

Example:

The following rule applies to dividing two powers: Powers with the same base are divided by subtracting the exponents and keeping the base. As always, the calculation rule and two examples:

an : am = an-m

Examples:

• 25 : 23 = 25-3 = 22 = 4
• 43 : 42 = 43-2 = 41 = 4

This power law should not be confused with the following law: Powers with the same exponent are divided by dividing the bases and keeping the exponent:

an : bn = (a: b)n

Examples:

• 42 : 22 = (4 : 2)2 = 22 = 4
• 93 : 33 = (9 : 3)3 = 33 = 27

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