# 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:

- 10
^{2}= 10 · 10 - 10
^{3}= 10 · 10 · 10 - 10
^{4}= 10 · 10 · 10 · 10

As you can see, it says 10^{2} the number 10 twice there with a mark in between. At 10^{3} 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 a^{n} 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:

- 2
^{3}= 2 · 2 · 2 = 8 - 7
^{4}= 7 · 7 · 7 · 7 = 2401 - 3
^{5}= 3 · 3 · 3 · 3 · 3 = 243 - 8
^{4}= 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.

Show:**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:

- 1,2
^{4}= 1,2 · 1,2 · 1,2 · 1,2 = 2,0736 - 2,34
^{3}= 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:

**a ^{n} · A^{m} = a^{n + m}**

**Examples**:

- 2
^{5}· 2^{3}= 2^{5+3}= 2^{8}= 256 - 4
^{-3}· 4^{7}= 4^{-3+7}= 4^{4}= 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:

**a ^{n} · B^{n} = (a b)^{n}**

**Examples:**

- 5
^{3}· 2^{3}= (5 · 2)^{3}= 10^{3}= 1000 - 3
^{5}· 2^{5}= (3 · 2)^{5}= 6^{5}= 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:

**(a ^{n})^{m} = a^{n · 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:

**a ^{n} : a^{m} = a^{n-m}**

**Examples:**

- 2
^{5}: 2^{3}= 2^{5-3}= 2^{2}= 4 - 4
^{3}: 4^{2}= 4^{3-2}= 4^{1}= 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:

**a ^{n} : b^{n} = (a: b)^{n}**

**Examples:**

- 4
^{2}: 2^{2}= (4 : 2)^{2}= 2^{2}= 4 - 9
^{3}: 3^{3}= (9 : 3)^{3}= 3^{3}= 27

**Left:**

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