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Index Laws Question Generator

Simplify an expression like 4x⁵ × 3x² ÷ 6x³. The mistake this drills out is treating the coefficients and the indices the same way: the numbers multiply and divide, the indices add and subtract, and mixing the two is the single most common error in the topic. Questions are constructed so the simplified coefficient is always a whole number and the simplified power is always positive, because a fractional coefficient turns an index question into a fractions question. At the stretch level a second letter joins in, so the powers of x and the powers of y have to be tracked apart from each other as well as from the numbers.

What this generator does

Builds a chain of two or three power terms joined by multiplication and division, and asks for it as a single term. The working states the coefficient result and the index sum for each letter separately, which is the distinction the topic turns on.

How to use this tool

  1. Choose how many expressions to simplify and at what level.
  2. Simplify each to a single term.
  3. Reveal the answers to check both the coefficient and the power.
  4. Show the working to see the numbers and the indices treated separately.

Understanding the controls

How many questions
Between 1 and 20. These are short enough that twenty fits comfortably on a single side.
Level
Gentle uses two terms with no coefficients at all, so only the index law is exercised. Standard adds coefficients and a dividing term. Stretch raises the indices and brings in a second letter, so x and y powers have to be collected separately.
Seed
Gives the same expressions again from the same word.
Show the answers / working
The single simplified term, or the coefficient and index arithmetic behind it.

Common use cases

  • Drilling the add-for-multiply, subtract-for-divide rules
  • Separating the coefficient arithmetic from the index arithmetic
  • Quick starters before moving on to negative and fractional indices
  • Practice questions where the answer is always tidy
  • Moving on to expressions in two letters, where x and y powers must be kept apart
  • Giving a group the identical set from a shared seed

How this generator works

Powers are drawn for each letter in each term, with the last term dividing rather than multiplying above the gentle level. The dividing term’s coefficient is planted inside the first term, so the division comes out whole by construction rather than by luck, and any draw that would leave a power of zero or less is topped up before it is shown. The check then reads the printed question and the printed answer back as text and works the simplification out again from those characters alone, so a question whose answer does not follow from what it prints cannot be shown.

Randomness and fairness

The terms are drawn at random; the guarantee of a whole coefficient and a positive power is enforced, not lucky. Seeded sets reproduce exactly and are therefore explicitly not cryptographically secure. Unseeded, draws come from the browser's cryptographically secure generator.

For how randomness is produced across the whole site, see how Generate Random works.

Limitations and good to know

  • Two letters at most, x and y. Expressions in three or more letters are not generated.
  • Indices stay positive whole numbers, so negative and fractional indices are outside what this covers.
  • Brackets raised to a power — the (xᵃ)ᵇ law — are not among the forms produced.
  • Answers are revealed rather than marked.
  • Nothing is stored between visits; seed a set you want again.

Privacy and your data

Expressions are assembled in your browser. Your seed and settings are not transmitted and are not remembered after you leave.

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