Exponential Expressions

In this lesson, you will be learning what an exponent is and about the properties and rules of exponents. You will also learn how to use exponents in problem solving.

Definition: An exponent is a power of a number that shows how many times that number is multiplied by itself.

An example would be \(\begin{align*}2^3\end{align*}\). You would multiply 2 by itself 3 times, \(\begin{align*}2 \times 2 \times 2\end{align*}\). The number 2 is the base and the number 3 is the exponent. The value \(\begin{align*}2^3\end{align*}\) is called the power.

Example 1: Write in exponential form: \(\begin{align*}\alpha \times \alpha \times \alpha \times \alpha\end{align*}\).

Solution: You must count the number of times the base, \(\begin{align*}\alpha\end{align*}\), is being multiplied by itself. It's being multiplied four times so the solution is \(\begin{align*}\alpha^4\end{align*}\).

Note: There are specific rules you must remember when taking powers of negative numbers.

\(\begin{align*}(\text{negative number)} \times (\text{positive number}) &= \text{negative number}\\ (\text{negative number}) \times (\text{negative number}) &= \text{positive number}\end{align*} \)

For even powers of negative numbers, the answer will always be positive. Pairs can be made with each number and the negatives will be cancelled out.

\(\begin{align*}(-2)^4 = (-2)(-2)(-2)(-2) = (-2)(-2) \cdot (-2)(-2) = +16\end{align*}\)

For odd powers of negative numbers, the answer is always negative. Pairs can be made but there will still be one negative number unpaired, making the answer negative.

\(\begin{align*}(-2)^5 = (-2)(-2)(-2)(-2)(-2) = (-2)(-2) \cdot (-2)(-2) \cdot (-2) = -32\end{align*}\)

When we multiply the same numbers, each with different powers, it is easier to combine them before solving. This is why we use the Product of Powers Property.

Product of Powers Property: For all real numbers \(\begin{align*}\chi, \chi^n \cdot \chi^m = \chi^{n+m}\end{align*}\).

Example 2: Multiply \(\begin{align*}\chi^4 \cdot \chi^5\end{align*}\).

Solution: \(\begin{align*}\chi^4 \cdot \chi^5 = \chi^{4+5} = \chi^9\end{align*}\)

Note that when you use the product rule you DO NOT MULTIPLY BASES.

Example: \(\begin{align*}2^2 \cdot 2^3 \neq 4^5\end{align*}\)

Another note is that this rule APPLIES ONLY TO TERMS THAT HAVE THE SAME BASE.

Example: 

\(\begin{align*}2^2 \cdot 3^3 \neq 6^5\end{align*}\)

\(\begin{align*}& (x^4)^3 = x^4 \cdot x^4 \cdot x^4 \qquad \qquad 3 \ \text{factors of} \ x \ \text{to the power} \ 4.\\ & \underbrace{(x \cdot x \cdot x \cdot x}_{x^4}) \cdot \underbrace{(x \cdot x \cdot x \cdot x}_{x^4}) \cdot \underbrace{(x \cdot x \cdot x \cdot x}_{x^4})=\underbrace{(x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x)}_{x^{12}}\end{align*}\)

This situation is summarized below.

Power of a Product Property: For all real numbers \(\begin{align*}\chi\end{align*}\),

\(\begin{align*}(\chi^n)^m = \chi^{n \cdot m}\end{align*}\)

The Power of a Product Property is similar to the Distributive Property. Everything inside the parentheses must be taken to the power outside. For example, \(\begin{align*}(x^2y)^4=(x^2)^4 \cdot (y)^4=x^8y^4\end{align*}\). Watch how it works the long way.

\(\begin{align*}\underbrace{(x \cdot x \cdot y)}_{x^2y} \cdot \underbrace{(x \cdot x \cdot y)}_{x^2y} \cdot \underbrace{(x \cdot x \cdot y)}_{x^2y} \cdot \underbrace{(x \cdot x \cdot y)}_{x^2y}=\underbrace{(x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot y \cdot y \cdot y \cdot y)}_{x^8y4}\end{align*}\)

The Power of a Product Property does not work if you have a sum or difference inside the parenthesis. For example, \(\begin{align*}(\chi+\gamma)^2 \neq \chi^2 + \gamma^2\end{align*}\). Because it is an addition equation, it should look like \(\begin{align*}(\chi+\gamma)(\chi+\gamma)\end{align*}\).

Example 3: Simplify \(\begin{align*}(\chi^2)^3\end{align*}\).

Solution: \(\begin{align*}(\chi^2)^3= \chi^{2\cdot 3} = \chi^6\end{align*}\)

Source: cK-12, https://www.ck12.org/book/basic-algebra/section/8.1/
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