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Index Law (Multiplication) Calculator

Rule for multiplying terms with the same base.

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Resulting Index

Formula first

Overview

The Multiplication Index Law states that when two powers with the same base are multiplied, their exponents are added together to simplify the expression. This principle is a fundamental rule of algebra derived from the repetitive nature of multiplication in exponentiation.

Symbols

Variables

m = First Index, n = Second Index, m+n = Resulting Index

First Index
Variable
Second Index
Variable
m+n
Resulting Index
Variable

Apply it well

When To Use

When to use: Apply this law whenever you are multiplying terms that share an identical base. It is essential to ensure the bases are exactly the same, as the rule does not apply to terms with different bases.

Why it matters: This rule allows for the rapid simplification of complex algebraic expressions and is foundational for scientific notation. It is used extensively in calculus, physics, and engineering to manage scale and solve growth-related equations.

Avoid these traps

Common Mistakes

  • Multiplying the indices instead of adding them.
  • 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 x⁵ × x³. What is the resulting exponent (result_index)?

First Index5
Second Index3

Solve for:

Hint: Add the two exponents together because the bases are the same.

Practice Problem 2

In the equation x⁷ ×xⁿ = x¹², solve for the missing exponent n.

First Index7
Resulting Index12

Solve for:

Hint: Subtract the known exponent from the total resulting exponent.

Practice Problem 3

An expression x⁻³ is multiplied by xⁿ to give a final result of x¹⁵. Find the value of n.

First Index-3
Resulting Index15

Solve for:

Hint: Be careful with the negative sign; you are solving -3 + n = 15.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Wikipedia: Exponentiation
  2. Britannica: Exponentiation
  3. Wikipedia: Laws of exponents
  4. Atkins' Physical Chemistry
  5. Halliday, Resnick, and Walker, Fundamentals of Physics
  6. AQA GCSE Maths — Number (Indices)