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Multiplication of a power with the same base

If the same bases are given in the multiplication of two powers (or more), the result in the multiplication of the powers are equally large powers, in which the base number is the base a with the superscript of the sum of all superscript, that where given as the base a. This is what it would look like in numbers:

a^u · a^v = a^{u + v}

It is like this because, for example:

2⁷ · 2³ = 2¹⁰

2 ·2 · 2 · 2 · 2 · 2 · 2    ·    2 ·2 · 2 = 2¹⁰

128 · 8 = 1024

Division of powers with the same base

The base (a) to the power of the number (u) divided by the same base (a) to the power of another number (v) equals the base (a) to the power of the difference of both numbers (u and v). This is what it would look like:

It is like this because, for example:

2⁴ = 16

Multiplication of powers with the same superscript

When the product of two powers have the same superscript, the bases (a and b) are multiplied and have the power of the mutual superscript (u).

a^u · b^u = (a · b)^u

Example:

2³ · 3³ = (2 · 3)³

8 · 27 = 216

Division of powers with the same superscript

The same counts, like the multiplication of powers with the same exponent. A number a to the power of the exponent u through another number to the power of u is the same thing, as the quotient of a and b to the power of the exponent u.

Example:

3,375 = 3,375

Raising to the power of powers

To take a power to the power of an exponent is the same thing as multiplying the exponents and taking the base to the power of the product.

(a^u)^v = a^{u · v}

Example:

(2²)⁵ = 2^{2 · 5}

(2 · 2)⁵ = 2 ·2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2

(2 · 2) (2 · 2) (2 · 2) (2 · 2) (2 · 2) = 2 ·2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2

2 ·2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 = 2 ·2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2

1024 = 1024

Copyright © 2005 Christian Franzki - Emkacom Group
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02/09/07