MathematicsNumber and AlgebraGCSE
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Index Law (Division) Calculator

Rule for dividing terms with the same base.

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Resulting Index

Formula first

Overview

The Quotient Rule for Indices states that when dividing powers with the same base, the exponent of the divisor is subtracted from the exponent of the dividend. This principle simplifies the division of exponential terms by converting the operation into a basic subtraction of powers.

Symbols

Variables

m = Top Index, n = Bottom Index, m-n = Resulting Index

Top Index
Variable
Bottom Index
Variable
m-n
Resulting Index
Variable

Apply it well

When To Use

When to use: Use this law whenever you are dividing two terms that share the same base, regardless of whether the base is a constant or a variable. It applies to all real number exponents, including positive, negative, and fractional values.

Why it matters: This law is a fundamental algebraic shortcut that allows for the rapid simplification of complex mathematical models in fields like engineering and financial analysis. It is essential for understanding more advanced topics such as calculus and scientific notation.

Avoid these traps

Common Mistakes

  • Dividing the indices instead of subtracting 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

A student is asked to simplify the algebraic expression x⁸ ÷ x³. If the final answer is written in the form x to the power of result_index, what is the value of result_index?

Top Index8
Bottom Index3

Solve for:

Hint: Subtract the exponent of the divisor (n) from the exponent of the dividend (m).

Practice Problem 2

Simplify the expression x⁵ / x⁻² and determine the resulting power of the base x.

Top Index5
Bottom Index-2

Solve for:

Hint: Subtracting a negative number is the same as adding the positive version of that number.

Practice Problem 3

In a physics calculation, a variable xᵐ is divided by x⁴, resulting in the term x⁶. Calculate the value of the initial exponent m.

Bottom Index4
Resulting Index6

Solve for:

Hint: Rearrange the subtraction formula to solve for the missing dividend exponent: m = result_index + n.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Wikipedia: Exponentiation
  2. Britannica: Exponent
  3. Britannica: Exponentiation
  4. AQA GCSE Maths — Number (Indices)