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Concept Version 4
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Rules for Exponent Arithmetic

There are rules for operating on numbers with exponents that make it easy to simplify and solve problems.

Learning Objective

  • Explain and implement the rules for operating on numbers with exponents


Key Points

    • The rule $a^m \cdot a^n = a^{m+n}$ applies when multiplying two exponential expressions with the same base, provided the base is a non-zero integer.
    • The rule $\displaystyle \frac{{a}^{m}}{{a}^{n}}={a}^{m-n}$ applies when dividing two exponential expressions with the same base, provided the base is a non-zero integer.
    • The rule ${({a}^{n})}^{m}={a}^{n \cdot m}$ applies when raising an exponential expression to another exponent for any non-zero integer $a$.
    • The rule ${(ab)}^{n}={a}^{n}{b}^{n}$applies when raising a product to an exponent for any non-zero integers $a$ and $b$.

Terms

  • base

    In an exponential expression, the value that is multiplied by itself.

  • exponent

    In an exponential expression, the value raised above the base; represents the number of times the base must be multiplied by itself.


Full Text

There are several useful rules for operating on numbers with exponents. The following four rules, also known as "identities," hold for all integer exponents, provided that the base is non-zero.

Multiplying Exponential Expressions with the Same Base  

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

$a^m$ means that you have $m$ factors of $a$. If you multiply this quantity by $a^n$, i.e. by $n$ additional factors of $a$, then you have $a^{m+n}$ factors in total. For example:

$5^3 \cdot 5^4=5^{3+4}=5^7$

Note that you can only add exponents in this way if the corresponding terms have the same base.

Dividing Exponential Expressions with the Same Base

$\displaystyle \frac{{a}^{m}}{{a}^{n}}={a}^{m-n}$

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

In order to visualize this process, consider the fraction:

$\dfrac{3^5}{3^2}$

This fraction can be rewritten as:

$\dfrac{3 \cdot 3 \cdot 3 \cdot 3 \cdot 3}{3 \cdot 3}$

Here you can see that two 3s will cancel out from the numerator and denominator. We are left with:

$\dfrac{3 \cdot 3 \cdot 3}{1} = 3^3$

As an additional example:

$\displaystyle \frac{7^4}{7^2}=7^{4-2}=7^2$

Raising an Exponential Expression to an Exponent

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

If you think about an exponent as telling you that you have a certain number of factors of the base, then ${({a}^{n})}^{m}$ means that you have factors $m$ of $a^n$. Therefore, you have $m$ groups of $a^n$, and each one of those has $n$ groups of $a$. Therefore, you have $m$ groups of $n$ groups of $a$; therefore, you have $n \cdot m$ groups of $a$, or ${a}^{n \cdot m}$. For example:

$(3^3)^3=3^{3 \cdot 3}=3^9$

Raising a Product to an Exponent

${(ab)}^{n}={a}^{n}{b}^{n}$

You can multiply numbers in any order you please. Instead of multiplying together $n$ factors equal to $ab$, you could multiply all of the $a$s together and all of the $b$s together and then finish by multiplying $a^n$ by $b^n$. For example:

$(4\cdot5)^3=4^3\cdot5^3$

Example

Simplify the following expression: $(3\cdot2)^3\cdot (2^5)^4$

For the first part of the expression, apply the rule for a product raised to an exponent:

$(3\cdot2)^3\cdot (2^5)^4 = 3^3 \cdot 2^3 \cdot (2^5)^4$

For the last part of the expression, apply the rule for raising an exponential expression to an exponent:

$3^3 \cdot 2^3 \cdot (2^5)^4 = 3^3 \cdot 2^3 \cdot 2^{5\cdot 4} = 3^3 \cdot 2^3 \cdot 2^{20}$

Notice that two of the terms in this expression have the same base: 2. These two terms can be combined by applying the rule for multiplying exponential expressions with the same base:

$3^3 \cdot 2^3 \cdot 2^{20} = 3^3 \cdot 2^{3+20} = 3^3 \cdot 2^{23}$

Therefore, $3^3 \cdot 2^{23}$ is the simplified form of this expression.

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