Exponents: Basic Rules
By the end of this lesson, you’ll be able to:
- Apply the core exponent rules for multiplication, division, and powers.
- Evaluate expressions with positive integer exponents.
- Distinguish clearly between bases and exponents.
- Simplify expressions by identifying which rule applies to each situation.
Key Ideas
An exponent tells how many times a base is multiplied by itself.
Example:
\[
3^4 = 3 \cdot 3 \cdot 3 \cdot 3 = 81
\]
Exponent Rules (Summary Table)
| Rule | Description | Formula |
|---|---|---|
| Identity Rule | Any base to the first power equals itself | \[a^1 = a\] |
| Product Rule | Multiply powers with the same base | \[a^m \cdot a^n = a^{m+n}\] |
| Quotient Rule | Divide powers with the same base | \[\frac{a^m}{a^n} = a^{m-n}\] |
| Power Rule | Raise a power to another power | \[(a^m)^n = a^{mn}\] |
| Zero Exponent | Any nonzero base raised to 0 equals 1 | \[a^0 = 1 \quad (a \ne 0)\] |
| Negative Exponent | A negative exponent creates a reciprocal | \[a^{-m} = \frac{1}{a^m}\] |
| Power of a Product | Power applies to each factor | \[(ab)^m = a^m b^m\] |
| Power of a Quotient | Power applies to numerator & denominator | \[\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m} \quad (b \ne 0)\] |
Exponents apply only to the base they touch, unless parentheses change the base.
Example: \(-3^2\) is different from \((-3)^2\).
Common Problem Types
1. Applying the Product Rule
Example:
\[
x^3 \cdot x^5 = x^{3+5} = x^8
\]
2. Applying the Quotient Rule
Example:
\[
\frac{y^7}{y^2} = y^{7-2} = y^5
\]
3. Power Rule
Example:
\[
(2^3)^2 = 2^{3 \cdot 2} = 2^6 = 64
\]
4. Zero Exponent
Example:
\[
(5x)^0 = 1
\]
5. Power of a Product
Example:
\[
(3xy)^2 = 3^2 x^2 y^2 = 9x^2 y^2
\]
Strategies
- Combine powers only when the bases match.
- Multiply exponents only when using the power rule.
- Use parentheses to clarify what the exponent actually applies to.
- Keep coefficients separate from variable exponents when simplifying.
- Expand small examples to check your understanding if unsure.
Worked Examples
Example 1
Simplify:
\[
a^4 \cdot a^6
\]
Solution:
- Bases match → use product rule.
- Add exponents → \(a^{4+6}\).
Answer: \(a^{10}\)
Example 2
Simplify:
\[
\frac{x^9}{x^3}
\]
Solution:
- Bases match → use quotient rule.
- Subtract exponents → \(x^{9-3}\).
Answer: \(x^6\)
Example 3
Simplify:
\[
(5^2)^3
\]
Solution:
- Use power rule → multiply exponents: \(2 \cdot 3 = 6\).
- Rewrite: \(5^6\).
Answer: \(5^6\)
Example 4
Simplify:
\[
(2x)^3
\]
Solution:
- Cube each factor: \(2^3\) and \(x^3\).
- \(2^3 = 8\).
Answer: \(8x^3\)
- Combining the bases instead of exponents (e.g., \(2^3 \cdot 2^4 \ne 4^7\)).
- Forgetting to apply the exponent to every factor inside parentheses.
- Misinterpreting zero exponent: \(a^0 = 1\) (not 0).
- Confusing \(-3^2\) with \((-3)^2\) due to missing parentheses.
- Applying exponent rules when the bases are different.
Practice Problems
- \(x^5 \cdot x^2\)
- \((3a)^2\)
- \(\frac{y^7}{y^4}\)
- \((t^3)^2\)
- \((4mn)^3\)
1. \(x^5 \cdot x^2\)
Add exponents → \(x^{7}\)
Answer: \(x^7\)
2. \((3a)^2\)
Square each factor → \(3^2 a^2 = 9a^2\)
Answer: \(9a^2\)
3. \(\frac{y^7}{y^4}\)
Subtract exponents → \(y^3\)
Answer: \(y^3\)
4. \((t^3)^2\)
Multiply exponents → \(t^6\)
Answer: \(t^6\)
5. \((4mn)^3\)
Cube each factor → \(4^3 m^3 n^3 = 64m^3 n^3\)
Answer: \(64m^3 n^3\)
Summary
- Product rule → add exponents.
- Quotient rule → subtract exponents.
- Power rule → multiply exponents.
- Zero exponent always gives 1 (except \(0^0\)).
- Negative exponent → reciprocal.
- Parentheses determine the base—identify them carefully.
- Expand simple examples to check which rule applies.
- Use parentheses to avoid confusion with negative bases.
- Keep coefficients separate from variable exponents while simplifying.