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Index Law (Negative Indices) Calculator

Rule for handling negative indices.

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Resulting Value

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Overview

The negative index law defines that a base raised to a negative power is equivalent to the reciprocal of that base raised to the corresponding positive power. This principle ensures mathematical consistency when subtracting exponents during division, even when the divisor's power exceeds the dividend's power.

Symbols

Variables

x = Base, n = Negative Index (numerical magnitude), = Resulting Value

Base
Variable
Negative Index (numerical magnitude)
Variable
Resulting Value
Variable

Apply it well

When To Use

When to use: Apply this law when simplifying algebraic terms containing negative exponents or when shifting variables between the numerator and denominator of a fraction. It is essential for converting small decimals into scientific notation or simplifying expressions before differentiation in calculus.

Why it matters: This law allows for a unified system of arithmetic that handles both growth and decay using the same notation. It is critical in scientific fields for representing microscopic measurements, such as the mass of an atom or the wavelength of light.

Avoid these traps

Common Mistakes

  • Thinking the result is a negative number.
  • 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

Evaluate the expression 5⁻² and provide the decimal result.

Base5
Negative Index (numerical magnitude)2

Solve for: result

Hint: Rewrite the expression as a fraction with a positive power in the denominator.

Practice Problem 2

If 2⁻ⁿ = 0.125, find the value of n.

Base2
Resulting Value0.125

Solve for:

Hint: Convert the decimal 0.125 into a fraction and then into a power of 2.

Practice Problem 3

Solve for x if x⁻³ = 0.015625 (which is equivalent to 1/64).

Negative Index (numerical magnitude)3
Resulting Value0.015625

Solve for:

Hint: Express the equation as 1/x³ = 1/64 and find the cube root.

The full worked solution stays in the interactive walkthrough.

References

Sources

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