Rules for Exponential Expressions

Recall the rules for operating on numbers with exponents, which are used when simplifying and solving problems in mathematics:

• Multiplying exponential expressions with the same base: $a^m \cdot a^n = a^{m+n}$
• Dividing exponential expressions with the same base: $\displaystyle \frac{{a}^{m}}{{a}^{n}}={a}^{m-n}$
• Raising an exponential expression to an exponent: ${({a}^{n})}^{m}={a}^{n \cdot m}$
• Raising a product to an exponent: ${(ab)}^{n}={a}^{n}{b}^{n}$

Previously, we have applied these rules only to expressions involving integers. However, they also apply to expressions involving a combination of both integers and variables. This makes them more broadly applicable in solving mathematics problems.

Exponential Expressions with Variables

In terms of conducting operations, exponential expressions that contain variables are treated just as though they are composed of integers. For example, consider the rule for multiplying two numbers with exponents. We know that $(2 \cdot 5)^2 = 2^2 \cdot 5^2 $. The same rule applies to expressions with variables. The following statements therefore hold true:

• $(4a)^3 = 4^3 \cdot a^3$
• $(xy)^2 = x^2y^2$

Each of the other rules for operating on numbers applies to expressions with variables as well. You will see how each of these applies in the following examples.

Example 1

Simplify the following expression:

$\displaystyle \frac{4a^7}{a^2}$

Now apply the rule for dividing exponential expressions with the same base:

$\displaystyle \frac{4a^7}{a^2} = 4a^{\left(7-2\right)} = 4a^5$

Example 2

Simplify the following expression:

$(a^3)(a^2) + (2b^2)^3$

To simplify the first part of the expression, apply the rule for multiplying two exponential expressions with the same base:

$(a^3)(a^2) = a^5$

To simplify the second part of the expression, apply the rule for multiplying numbers with exponents:

$(2b^2)^3 = (2)^3 \cdot (b^2)^3 $

Now, since we know that $2^3 = 2 \cdot 2 \cdot 2 = 8$, we can plug that in. We can also apply the rule for raising a power to another exponent:

$(2b^2)^3 = 8b^6$

Combining the two terms, our original expression simplifies to:

$a^5 + 8b^6$