Exponents & Polynomials
Exponent Rules
Applying the product, quotient, and power rules for exponents.
Exponent Rules with Variables and Products
Before reading on, predict what (2x)³ actually equals. Does the exponent 3 apply only to x, or to the entire quantity 2x — including the coefficient?
Since (2x)³ means (2x) multiplied by itself 3 times: (2x)(2x)(2x). Regrouping the factors — three 2's together and three x's together — gives 2³ · x³ = 8x³. The exponent applies to everything inside the parentheses, coefficient included, which is exactly the power-of-a-product rule.
Definition — Power of a Product and Power of a Quotient
Worked Example — Applying the Power of a Product Rule
Worked Example — Combining Multiple Exponent Rules
Worked Example — Applying the Power of a Quotient Rule
Tip
Common Mistakes
Applying an exponent to only the variable and skipping the coefficient, such as simplifying (3x²)³ as 3x⁶ instead of 27x⁶.
An exponent outside a parenthesis distributes to every factor inside, including any numeric coefficient — cube the 3 as well as the x².
Adding exponents when applying the power-of-a-power rule instead of multiplying them, such as evaluating (x³)² as x⁵.
The power rule multiplies exponents: (x³)² = x³ˣ² = x⁶ — reserve adding exponents for the product rule (same base multiplied together), a different situation entirely.
Key Takeaways
- Raising a product or quotient to a power distributes that power across every factor.
- Combining several exponent rules in one expression means applying each rule in turn, carefully tracking coefficients and variables separately.
- Every exponent rule ultimately traces back to counting repeated factors.
Summary
These exponent rules extend cleanly to expressions with coefficients, products, and quotients. The next lesson completes the exponent toolkit by handling negative and zero exponents within more complex expressions.
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