RWM102 Study Guide

Unit 7: Operations with Monomials

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:


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: 


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