A factor in the denominator can be moved to the numerator by changing the sign of its exponent.
When multiplying like bases, add the exponents.
Any nonzero expression raised to the zero power equals 1.
Final answers should be written without negative exponents.
💡 Main Idea
When an exponential expression is being multiplied by a fraction, you can rewrite the denominator factors as numerator factors with opposite exponents. Then all the terms are being multiplied. Now combine like bases by adding exponents. If any exponent becomes negative in the final result, move that factor across the fraction bar so the answer has no negative powers.
✏️ Worked Examples – Negative, Zero, Product, and Quotient Exponent Rules
Example 1 – Combine Like Bases
\(\displaystyle 4a^{-3}b^2\cdot\frac{2a^5}{b^3}\)
Step 1: Move the denominator factor to the numerator by changing the sign of its exponent.
\(4a^{-3}b^2\cdot2a^5b^{-3}\)
Step 2: Multiply the coefficients and combine powers with the same base.
\(8a^{-3+5}b^{2+(-3)}\)
\(8a^2b^{-1}\)
Step 3: Rewrite the expression using only positive exponents.
\(\displaystyle \frac{8a^2}{b}\)
Final Answer:
\(\displaystyle \boxed{\frac{8a^2}{b}}\)
Example 2 – Apply the Zero Exponent Rule
\(\displaystyle 3m^{-2}n^0\cdot\frac{5m^4}{n^2}\)
Step 1: Use the zero exponent rule.
\(n^0=1\)
So the expression becomes:
\(\displaystyle 3m^{-2}\cdot\frac{5m^4}{n^2}\)
Step 2: Move the denominator factor to the numerator by changing the sign of its exponent.
\(3m^{-2}\cdot5m^4n^{-2}\)
Step 3: Multiply the coefficients and combine like bases.
\(15m^{-2+4}n^{-2}\)
\(15m^2n^{-2}\)
Step 4: Rewrite the expression using only positive exponents.
Step 4: Rewrite the expression using only positive exponents.
\(\displaystyle \frac{2}{x^2y^2}\)
Final Answer:
\(\displaystyle \boxed{\frac{2}{x^2y^2}}\)
🧠 Math Vocabulary
Coefficient: The numerical factor in a term.
Exponent: The small raised number that tells how many times a base is multiplied by itself.
Base: The repeated factor in an exponential expression.
Negative Exponent: An exponent that indicates a factor belongs on the opposite side of the fraction bar.
Zero Exponent: Any nonzero base raised to the zero power equals 1.
📚 What You Should Already Know
Students should already know how to multiply coefficients, combine like bases when multiplying, and understand basic exponent notation.
🚀 What Comes Next
Next, students can apply these same exponent rules when dividing powers, simplifying rational expressions, and working with scientific notation and exponential equations.
🧩 Embed This Video in Your LMS
Teachers can embed this YouTube video directly into Canvas, Google Classroom, Schoology, or another learning platform. Click below to show the embed code.
Copy and paste this code into your LMS to embed the video.
