Integer
Exponents
Master the properties of integer exponents to simplify and generate equivalent numerical expressions with confidence.
8th Grade Math
What Is an Exponent?
An exponent tells you how many times to multiply a number (the base) by itself. The expression \( a^n \) means: multiply \( a \) by itself \( n \) times.
Zero and Negative Exponents
Exponents aren't limited to positive whole numbers. Two critical definitions extend the idea:
Zero Exponent
Any nonzero base raised to the zero power equals 1: \( a^0 = 1 \) (where \( a \neq 0 \)).
Negative Exponent
A negative exponent means take the reciprocal: \( a^{-n} = \dfrac{1}{a^n} \).
Why Do Exponents Matter?
Exponent rules let you simplify complex expressions quickly. They appear in science (scientific notation), finance (compound interest), and throughout algebra.
The Five Core Rules
1
Product Rule: \( a^m \cdot a^n = a^{m+n} \)
2
Quotient Rule: \( \dfrac{a^m}{a^n} = a^{m-n} \)
3
Power of a Power: \( (a^m)^n = a^{m \cdot n} \)
4
Power of a Product: \( (ab)^n = a^n \cdot b^n \)
5
Power of a Quotient: \( \left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n} \)
Product Rule: Add the Exponents
Quotient Rule: Subtract the Exponents
Power of a Power: Multiply Exponents
Product vs Quotient Power Rules
| Rule | Power of a Product | Power of a Quotient |
|---|---|---|
| Formula | \( (ab)^n = a^n \cdot b^n \) | \( \left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n} \) |
| Idea | Distribute exponent to each factor | Distribute exponent to top and bottom |
| Example | \( (2 \cdot 5)^3 = 2^3 \cdot 5^3 = 1000 \) | \( \left(\dfrac{3}{4}\right)^2 = \dfrac{9}{16} \) |
When to Add vs Multiply Exponents
Add Exponents
When you multiply same-base powers: \( a^m \cdot a^n = a^{m+n} \). The bases stay, exponents add.
Multiply Exponents
When you raise a power to a power: \( (a^m)^n = a^{mn} \). Nest the operations, multiply the counts.
How to Simplify Step by Step
1
Identify the bases. Group terms that share the same base together.
2
Choose the right rule. Multiplying same base? Add. Dividing? Subtract. Power of a power? Multiply.
3
Apply the rule to combine or simplify exponents into a single power.
4
Handle negatives. Rewrite any negative exponent as a reciprocal: \( a^{-n} = \dfrac{1}{a^n} \).
5
Compute the value if a numerical answer is needed.
Example 1: Product Rule
Simplify \( 3^2 \times 3^{-5} \).
Example 2: Power of a Power
Simplify \( (2^4)^{-2} \).
Example 3: Multi-Step Simplification
Simplify \( \dfrac{(5^3)^2 \cdot 5^{-4}}{5^2} \).
Common Mistakes
✕ Multiplying Instead of Adding
Writing \( a^m \cdot a^n = a^{mn} \) instead of \( a^{m+n} \). Product rule means add the exponents.
✕ Negative Exponent = Negative Number
Thinking \( 2^{-3} = -8 \). Actually, \( 2^{-3} = \dfrac{1}{8} \). Negative exponent means reciprocal, not negative value.
✕ Distributing Over Addition
Writing \( (a + b)^2 = a^2 + b^2 \). Exponents only distribute over multiplication and division, not addition.
Quick Reference Table
| Property | Rule | Example |
|---|---|---|
| Product | \( a^m \cdot a^n = a^{m+n} \) | \( 4^2 \cdot 4^3 = 4^5 \) |
| Quotient | \( \dfrac{a^m}{a^n} = a^{m-n} \) | \( \dfrac{7^5}{7^2} = 7^3 \) |
| Power of Power | \( (a^m)^n = a^{mn} \) | \( (3^2)^4 = 3^8 \) |
| Zero Exponent | \( a^0 = 1 \) | \( 99^0 = 1 \) |
| Negative Exponent | \( a^{-n} = \dfrac{1}{a^n} \) | \( 6^{-2} = \dfrac{1}{36} \) |
Question 1 of 5
Simplify \( 7^3 \cdot 7^{-5} \).
Using the product rule: \( 7^3 \cdot 7^{-5} = 7^{3+(-5)} = 7^{-2} = \dfrac{1}{7^2} = \dfrac{1}{49} \).
Question 2 of 5
What is \( (4^2)^3 \) equal to?
Power of a power: multiply the exponents. \( (4^2)^3 = 4^{2 \cdot 3} = 4^6 = 4096 \).
Question 3 of 5
Simplify \( \dfrac{10^6}{10^{-2}} \).
Quotient rule: subtract exponents. \( \dfrac{10^6}{10^{-2}} = 10^{6-(-2)} = 10^{6+2} = 10^8 \). Subtracting a negative means adding.
Question 4 of 5
Which expression equals \( \dfrac{1}{81} \)?
Since \( 3^4 = 81 \), the reciprocal is \( \dfrac{1}{81} = 3^{-4} \). Note that \( 9^{-1} = \dfrac{1}{9} \) and \( 3^{-3} = \dfrac{1}{27} \), which are different values.
Question 5 of 5
Simplify \( \left(\dfrac{2}{5}\right)^{-3} \).
A negative exponent flips the fraction: \( \left(\dfrac{2}{5}\right)^{-3} = \left(\dfrac{5}{2}\right)^{3} = \dfrac{5^3}{2^3} = \dfrac{125}{8} \).