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Index Law (Fractional Indices) Calculator

Rule for handling fractional indices (roots).

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Resulting Value

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Overview

Fractional indices provide a notation to represent roots as exponents, where the denominator of the fraction indicates the degree of the root. This law bridges the gap between radical expressions and power operations, allowing for the application of standard algebraic index rules to roots.

Symbols

Variables

x = Base, n = Root Degree (denominator of fractional index), = Resulting Value

Base
Variable
Root Degree (denominator of fractional index)
Variable
Resulting Value
Variable

Apply it well

When To Use

When to use: Apply this law when converting radical signs into exponent form for simplification or calculus operations. It is particularly useful when multiplying or dividing roots with different bases or when solving equations involving nth roots.

Why it matters: This concept unifies the laws of indices, enabling a consistent mathematical framework for growth modeling, signal processing, and financial calculations. It allows complex root operations to be performed using simple addition and subtraction of fractions.

Avoid these traps

Common Mistakes

  • Dividing the base by the index.
  • 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 81 to the power of 1/4 to find the result.

Base81
Root Degree (denominator of fractional index)4

Solve for: result

Hint: Find a number that multiplied by itself four times equals 81.

Practice Problem 2

If the cube root of a number x is equal to 4, what is the value of x?

Root Degree (denominator of fractional index)3
Resulting Value4

Solve for:

Hint: To isolate x, raise both sides of the equation to the power of n.

Practice Problem 3

Find the value of the index n if the nth root of 10000 is 10.

Base10000
Resulting Value10

Solve for:

Hint: Identify how many times 10 must be multiplied by itself to reach 10000.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Britannica: Root (mathematics)
  2. Britannica: Exponentiation
  3. Wikipedia: Nth root
  4. Wikipedia: Exponentiation
  5. Wikipedia: Dimensional analysis
  6. Halliday, Resnick, and Walker, Fundamentals of Physics
  7. Atkins' Physical Chemistry
  8. AQA GCSE Maths — Number (Indices)