Algebra/Exponents

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Algebra
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[edit] Exponents

Why use exponents?

Exponents are a shortcut method used to write a number with the same factor repeated. Exponents are more effective with larger numbers but can be used with smaller ones as well. Exponents are written in superscript after a regular-sized number. The number in larger font is called the base. The number in superscript is the exponent and describes how many times the base is multiplied by itself.

For example:

The number 8 can be written as 23 because 2 x 2 x 2 = 8

In this example, 2 is the base and 3 is the exponent.

23 is read as "2 cubed" or "2 raised to the third power".

[edit] Basic Rules of Exponents

Multiplying Powers with the Same Base

a^{n} \cdot a^{m}=a^{n+m}

an mean that you have a a factor of n times. If you add m more factors of a then you have n + m factors of a.


Dividing Powers with the Same Base

\frac{a^{n}}{a^{m}}= \frac{\overbrace{a \cdot a \cdot a \cdot ... \cdot a}^\text{n factors of a}}{\underbrace{a \cdot a \cdot a \cdot ... \cdot a}_\text{m factors of a}}=a^{n-m}


In the same way that a^{n} \cdot a^{m}=a^{n+m} because you are adding on factors of a, dividing is taking away factors of a. If you have m factors of a in the denominator, then you can cross out m factors from the numerator. If there were n factors in the numerator now you have n-m factors in the numerator.


Raising a Power to a Power

(a^n)^{m}=a^{n \cdot m}

If you think about an exponent as telling your that you have so many factors of the base, then (an)m means that you have m factors of an. So you have m groups of an and each one of those has n groups of a. So you have m groups of n groups of a. So you have n \cdot m groups of a, or a^{n \cdot m}


Products raised to powers

(ab)n = anbn

You can multiply numbers in any order you please. Instead of muliplying together n factors equal to ab, you could multiply all of the a 's together, multiply all the b 's together, then finish by multiplying an times bn


Quotients raised to powers

\left( \frac{a}{b} \right)^{n}=\frac{a^n}{b^n}

You can raise a fraction to a power by raising the numerator to the power and the denominator to the power.


Any Nonzero number Raised to the Zero Power Is One

a \ne 0 \implies a^{0} = 1

This all means that as long as the base is not zero, when you have an exponent of zero, the expression is always equal to 1.


Proof:
\ \frac{a^n}{a^n} = a^{n - n} = a^0 = 1 \ ,\ a \ne 0


Negative Exponents

A negative sign on an exponent means that you need to take the reciprocal of the base, then the exponent becomes positive.

\left( \frac{a}{b} \right)^{-n}=\left( \frac{b}{a} \right)^{n}

[edit] Examples of Basic Rules of Exponents

Multiplying Powers with the Same Base
Add the exponents:

3^{4} \cdot 3^{2}= (3 \cdot 3\cdot 3\cdot3) \cdot (3\cdot3)=3 \cdot 3\cdot 3\cdot3 \cdot 3\cdot3=3^{6}


Dividing Powers with the Same Base
Subtract the exponents:

\frac{5^{7}}{5^{4}}= \frac{5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5}{5 \cdot 5 \cdot 5 \cdot 5}=\frac{5 \cdot 5 \cdot 5}{1}=5^{3}


Raising a Power to a Power
Multiply the exponents:

(4^{3})^{2}=4^{3} \cdot 4^{3} = (4 \cdot 4 \cdot 4) \cdot (4 \cdot 4 \cdot 4)=4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4=4^{6}



Any Nonzero number Raised to the Zero Power Is One

20 = 1

1090 = 1

− 920 = − 1

( − 92)0 = 1

[edit] Sites/Games for More Practice

http://www.shodor.org/interactivate/activities/OrderOfOperationsFou/ (order of operations including exponents)

http://jasperstreet.homeip.net/wiki/index.php/Math (everything I know about Maths)

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