RWM102 Study Guide

7a. Apply the rules of exponents to simplify algebraic exponential expressions

• What happens when you multiply two monomials?
• What does a negative exponent mean, and how can you change a negative exponent to a positive exponent?

Exponents are simply a shorter way to write repeated multiplication. Since $2^4=2 \times 2 \times 2 \times 2$, when we try to simplify $2^4 \times 2^3$, we need to remember this is four 2's multiplied with three 2's, meaning we have seven 2's multiplied together, or $2^7$. We can therefore see that multiplication property states: $a^m \times a^n = a ^ {m+n}$. Similarly, $(3x)^3 \times (3x)^7=(3x)^{10}$.

When dealing with exponents, you may come across a negative exponent. A negative exponent doesn't mean the answer will be negative. Instead, it means to take the reciprocal of the value, what you might call "flipping it". For example, $2^{-3}$ simply means the reciprocal of $2^3$, which is $\dfrac{1}{2^3}$. Similarly, if there is a negative exponent in the denominator of a fraction, it moves the term to the numerator. For example, $\dfrac{1}{3^{-2}}=3^2$.

To review, see:

• Exponential Expressions
• Negative Exponent Intuition
• Practice Solving Problems with Negative Exponents

7b. Multiply, divide, and simplify the powers of monomials

• How do you divide monomials with exponents?
• How do you take an exponent to another exponent?
• How do you multiply and divide different monomials?

When we divide monomials with exponents, we subtract our exponents, rather than adding, like we do when we multiply. For example, $\dfrac{4^5}{4^3}=4^2$. Therefore, the division property states: $\dfrac{a^m}{a^n}  = a^{m-n}$. Similarly, $\dfrac{x^3}{x^7}=x^{-4}$.

Sometimes, you might even have an exponent taken to another exponent, such as $(x^2)^3$. When this happens, you need to multiply the exponents, giving us $(x^2)^3=x^{2 \times 3}=x^6$. Therefore, the power property states: $(^am)^n=a^{m \times n}$. Similarly, $(3x^3)^4=3^4 x^{3 \times 4} =81x^{12}$.

We don't have to just multiply and divide the same monomial, we can multiply different monomials as well. To simplify the expression $(3x^2)(5x^3)$, we will multiply the numbers as normal, and then add the exponents on the variable, giving us $15x^5$.

Finally, we can divide different monomials. For example, $\dfrac{8x^4}{4x^3}$ can be simplified by first simplifying the numbers in the fraction, then using the division property to subtract the exponents, giving us an answer of $2x$.

To review, see: 

• Exponential Properties Involving Products
• Exponential Properties Involving Quotients
• Multiplication Properties of Exponents
• Dividing Monomials

Unit 7 Vocabulary

This vocabulary list includes terms listed above that students need to know to successfully complete the final exam for the course.

• exponents 
• multiplication property
• negative exponents
• reciprocal
• denominator 
• numerator 
• monomials
• division property
• power property