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Index Law (Power of a Power) Calculator

Rule for raising a power to another power.

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Resulting Index

Formula first

Overview

The Power of a Power rule states that when an exponential expression is raised to another exponent, the two exponents are multiplied while the base remains unchanged. This fundamental algebraic property simplifies nested powers by consolidating them into a single exponent.

Symbols

Variables

m = Inner Index, n = Outer Index, mn = Resulting Index

Inner Index
Variable
Outer Index
Variable
mn
Resulting Index
Variable

Apply it well

When To Use

When to use: Apply this law when an algebraic term already containing an exponent is enclosed in parentheses and raised to another power. It assumes the base is a real number and the exponents are rational, facilitating the simplification of complex polynomials.

Why it matters: This rule is crucial for calculating compound growth, dimensional scaling in geometry, and managing large numbers in scientific notation. It provides the mathematical foundation for logarithmic transformations and software algorithms that handle high-order calculations.

Avoid these traps

Common Mistakes

  • Adding the indices instead of multiplying.
  • Convert units and scales before substituting, especially percentages, time units, or powers of ten.
  • Interpret the answer with its unit and context; a percentage, rate, ratio, and physical quantity do not mean the same thing.

One free problem

Practice Problem

Practice Problem 1

Simplify the expression (x²)⁴. What is the value of the resulting index?

Inner Index2
Outer Index4

Solve for:

Hint: Multiply the inner exponent by the outer exponent.

Practice Problem 2

A growth factor is represented by the term (y⁵)³. Calculate the final exponent of y.

Inner Index5
Outer Index3

Solve for:

Hint: Recall that raising a power to a power involves multiplication of indices.

Practice Problem 3

If the expression ()⁶ is equivalent to z¹², find the value of the initial internal index m.

Outer Index6
Resulting Index12

Solve for:

Hint: Divide the final index by the outer exponent to find the missing factor.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Wikipedia: Exponentiation
  2. Stewart, Calculus: Early Transcendentals
  3. Britannica: Exponent
  4. Britannica: Exponentiation
  5. AQA GCSE Maths — Number (Indices)