# Speak Math Now!/Week 9: Six rules of Exponents

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 Educational level: this is a secondary education resource.
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## Rule 1 (Product of Powers)

am • an = am + n

Multiply exponents with the same base - add exponents

Examples

Here, we will list examples of this rule. If you have any questions on how some of these examples have been done, please go to the talk page.

• x • xxxx = x5
• b2 • b5 = b7

## Rule 2 (Power to a Power)

(am)n = am • n

Exponents with an exponent: multiply exponents.
-if you have multiple numbers in your parenthesis, all numbers get the exponent.

Examples
• (d5)3 = d15
• (xyz)2 = x2y2z2
• (xyz2)3 = x3y3z6

## Rule 3 (Multiple Power Rules)

As the title says, multiple power rules
Rule #1 + #2 in the same problem:

Examples
• 6(x3y)(x2y)3
• 6(x3y)(x6y3)
• 6x9y4

## Rule 4 (Quotient of Powers)

Divide numbers, subtract exponents

Examples
• am/an= am-n
• x4/x2=x2
• 6x7y3/12x3y8
• [divide 6 and 12 by "6", minus the exponents 7 and 3, minus the exponents 3 and 8].
• 1x4/2y5
• [clean up the 1--DON'T INCLUDE 1!]
• x4/2y5

## Rule 5 (Power of a Quotient)

An exponent affects all...

Examples
• (a/b)m = am/bm
• (2a3b5/3b2)2
• (22a6b10/32b4)
• [All numbers get affected, including exponents!]
• (4a6b10/9b4)
• [There are two "b"s, so you have to subtract b10 by b4... because the result is positive (b6), this number goes in the numerator space]
• (4a6b6/9)

## Rule 6 (Negative Exponents)

This does not apply to negative numbers, but exponents! As discussed in the next section, it is common to demand that the base of an exponential function is a positive number (that does not equal 1.)

• a-n = 1/an
• 1/a-n = an/1

• 4a-3b6/16x2b-2
• 4b2b6/16x2a3
• 4b8/16x2a3
• b8/4x2a3

### The base is best left positive

Review. To identify the three parts to an exponential expression, consider the case where B=2, X=3, and V=8:

${\displaystyle B^{X}=2^{3}=8=V}$

Here:

• B is the base of the exponential expression, with the requirements that B≠1 and 0<B<.
• X is the expression's exponent. While the consequences of letting X have an imaginary part are fascinating, this discussion considers only the case where X is a real number: −<X<.
• V is the value of the exponential expression.

2. It is worth mentioning why we exclude negative values of B from any discussion where we wish to all X to range over all positive and negative values on the real axis. It is well-known that for B=−1, B1/2=${\displaystyle {\sqrt {-1}}=i}$  is an imaginary number. It can also be shown that ${\displaystyle {\sqrt {-2}}}$  is also imaginary.[1]
1. ${\displaystyle {\sqrt {-2}}={\sqrt {2}}i}$  (Just Google it!)