Power of Product Rule Guide
The power of product rule is a key exponent law. It says a power outside a product applies to every factor inside. This means (ab)^n becomes a^n b^n. The same idea works with numbers, variables, and signed coefficients.
Why the Rule Works
A product raised to a power means repeated multiplication. For example, (xy)^3 means xy times xy times xy. Regrouping gives x times x times x, and y times y times y. That becomes x^3 y^3. The rule is only a shortcut for careful grouping.
Using Coefficients
Coefficients follow the same rule. If the expression is (2x)^4, then 2 is also raised to 4. The result is 16x^4. Signs matter too. A negative coefficient raised to an even integer gives a positive value. A negative coefficient raised to an odd integer stays negative.
Variable Exponents
Variables may already have powers. In (x^2 y^3)^4, each inner exponent is multiplied by 4. So x^2 becomes x^8, and y^3 becomes y^12. This uses the power of a power rule with the product rule.
Negative and Zero Powers
Negative exponents can stay in caret form, or move to the denominator. For example, x^-2 means 1 divided by x^2. A zero exponent usually becomes 1, when the base is not zero. Many teachers hide factors with zero powers.
Common Mistakes
Students often raise only the variable and forget the coefficient. They may also add exponents instead of multiplying by the outside power. Another mistake is dropping a negative sign too early. The best method is to write one step for each factor.
Practical Use
This calculator helps check algebra homework and build confidence. Enter a coefficient, variable factors, and an outside power. The result shows the expanded exponent steps. You can also export the answer for notes. Use it to compare manual work with a clear final expression.
Best Study Approach
Start with small products first. Then include coefficients and repeated variables. Check every factor line by line. Use positive powers before trying negative powers. When answers differ, compare each exponent step. This habit shows where the error starts. Over time, the rule feels natural, fast, and reliable during tests and quizzes under time limits.