Khan Academy offers a clear, step by step approach to the properties of exponents, helping learners connect integer exponents with algebraic expressions. These core rules underpin scientific notation, polynomial operations, and function analysis across higher math.
Free video lessons, interactive practice, and mastery challenges make exponent rules accessible whether you are refreshing skills or preparing for advanced coursework.
| Rule Name | Symbolic Form | Short Example | Key Condition |
|---|---|---|---|
| Product Rule | a^m * a^n = a^(m + n) | 2^3 * 2^4 = 2^7 | Same base |
| Quotient Rule | a^m / a^n = a^(m - n) | 5^6 / 5^2 = 5^4 | Nonzero base |
| Power of a Power | (a^m)^n = a^(m * n) | (7^2)^3 = 7^6 | Base remains unchanged |
| Power of a Product | (ab)^n = a^n * b^n | (3x)^2 = 3^2 * x^2 | Distribute exponent |
| Power of a Quotient | (a/b)^n = a^n / b^n | (4/y)^3 = 4^3 / y^3 | Nonzero denominator |
| Zero Exponent | a^0 = 1 | 11^0 = 1 | a ≠ 0 |
| Negative Exponent | a^(-n) = 1 / a^n | 2^(-3) = 1 / 2^3 | Nonzero base |
Product Rule for Exponents
When multiplying powers with the same base, add the exponents to simplify expressions efficiently.
How the Rule Works
Multiplying identical bases means repeated multiplication, so adding exponents reduces lengthy expansions into compact forms. This rule supports combining terms in equations and factoring polynomials.
Quotient Rule and Simplifying Fractions
Dividing like bases leads to subtraction of exponents, which is especially useful when simplifying algebraic fractions.
Handling Coefficients and Variables
Apply the quotient rule to variables and treat numerical coefficients separately using standard division, ensuring the base remains unchanged.
Power of a Power and Negative Exponents
Nested exponents multiply together, while negative exponents indicate reciprocals, enabling flexible rewriting of complex expressions.
Scientific Notation Applications
Converting very large or very small numbers into power of ten form relies on these rules for readability and computation.
Polynomial Operations and Exponent Properties
Mastery of exponent rules streamlines distribution, expansion, and factoring within polynomial equations.
Expanding and Simplifying Terms
Use the product and power rules to rewrite binomials raised to powers without expanding every multiplication manually.
Key Takeaways for Learners
- Add exponents when multiplying like bases.
- Subtract exponents when dividing like bases.
- Multiply exponents when raising a power to another power.
- Apply the exponent to every factor inside parentheses.
- Rewrite negative exponents as reciprocals to express answers with positive exponents.
- Verify that bases are identical before combining terms using these rules.
FAQ
Reader questions
What happens when the bases are different but the exponents are the same?
You cannot combine the bases directly using product or quotient rules; instead, group same exponents as (a * b)^n or evaluate numerically when possible.
Can you apply these rules to fractions with variables in the denominator?
Yes, rewrite the expression using negative exponents or apply the quotient rule, ensuring the denominator is not zero before simplifying.
How do you simplify an expression like (x^3 * y^2)^(-2)?
Distribute the negative exponent to each factor, resulting in x^(-6) * y^(-4), then rewrite with positive exponents as 1 / (x^6 * y^4).
What if one term in a product has a different base but the same exponent?
You can group them under a single exponent using the power of a product rule, writing (a * b)^n when both are raised to the same power n.