MA007 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?

We can also classify polynomials by their number of terms. For example, a single term, like \(2x^3\) is called a monomial. 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: \((a^m)^n=a^{m \times n}\). 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 you will need to know to successfully complete the final exam.

denominator
division property
exponent
monomial
multiplication property
negative exponent
numerator
power property
reciprocal