Algebra · step 2 of 13

Exponent & fraction rules

The index laws are short and unforgiving: aᵐ·aⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ, and a^(1/n) = ⁿ√a. Almost every error comes from inventing a law that does not exist — (a + b)² is not a² + b², and (a + b)⁻¹ is not a⁻¹ + b⁻¹. Powers distribute over products, never over sums.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Multiply — add the exponentsOnly when the bases match. x³·y³ does not combine as an exponent sum; it is (xy)³.
Divide — subtract the exponentsx⁵/x² = x³. Negative results are legitimate and give reciprocals.
Power of a power — multiply(x²)³ = x⁶. Distinguish this from x^(2³) = x⁸, where the tower associates upward.
Convert roots to fractional powersⁿ√(xᵐ) = x^(m/n). This turns every root manipulation into an index-law question.

Worked example

Simplify (8x⁶)^(2/3) / (2x²)².

Numerator: (8x⁶)^(2/3) = 8^(2/3)·x^(6·2/3) = 4x⁴, since 8^(1/3) = 2 and 2² = 4.
Denominator: (2x²)² = 4x⁴.
Divide: 4x⁴ / 4x⁴.
Coefficients cancel and x⁴/x⁴ = x⁰.

Answer. 1 (for x ≠ 0).

Where marks get dropped

These are the specific errors that cost credit on exponent & fraction rules questions — QED's rubric penalises each of them separately.

Writing (a + b)ⁿ = aⁿ + bⁿ. This fails for every n ≠ 1 — expand with the binomial theorem instead.
Applying a fractional power to only part of a product. (8x⁶)^(2/3) raises BOTH the 8 and the x⁶.
Treating a⁻ⁿ as a negative number. It is a reciprocal: 2⁻³ = 1/8, which is positive.

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Exponent & fraction rules — frequently asked questions

Why is a⁰ = 1?

Because aᵐ/aᵐ = a⁰ and any non-zero quantity divided by itself is 1. The case 0⁰ is left undefined or context-dependent for exactly this reason.

What does a^(m/n) mean?

The nth root of aᵐ, equivalently (ⁿ√a)ᵐ. Both readings agree for a > 0, which is why the fractional-index notation is safe there.

Can I take a fractional power of a negative number?

Only for odd denominators, and even then conventions vary — (−8)^(1/3) = −2 is standard, but (−8)^(2/6) is ambiguous. Most courses restrict fractional powers to non-negative bases.

The rest of Algebra

Foundational algebra to close prerequisite gaps. Each subtopic below has its own method, worked example and mark-losing traps.

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