Algebra 1/Unit 9: Six rules of Exponents

Rule 1 (Product of Powers)

a^m • aⁿ = a^{m + 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)

(a^m)ⁿ = a^{m • 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

• a^m/aⁿ= a^m-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 = a^m/b^m
• (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:

$B^X=2^3=8=V$

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.

Two comments are in order:

1. Though our choice of "X" and "V" as variables is not unusual, there is nothing "standard" about this notation. On the other hand, it is common to use the lower-case "b" to denote the base. The capital "B" was used here it because the lower-case "b" was used a number of times in the preceding examples.
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, B^1/2=$\sqrt{-1}=i$ is an imaginary number. It can also be shown that $\sqrt{-2}$ is also imaginary.