Index laws practice questions and worked solutions

An index is shorthand for repeated multiplication, so x⁵ means five copies of x multiplied together. Multiplying x⁵ by x³ therefore puts five copies alongside three copies, giving eight in total — which is why multiplying powers of the same base adds the indices rather than multiplying them.

Division cancels rather than accumulates. In x⁷ divided by x², two copies on top cancel with the two underneath and five remain, so the indices subtract. Rebuilding either law by writing out the copies takes fifteen seconds and settles any doubt about which operation applies.

The same-base condition is not decoration. There is no index law that simplifies x³ multiplied by y³ into a single power, because the copies are of different things and nothing cancels or combines. Questions mixing bases are testing whether students apply the laws mechanically or with understanding.

Raising a power to a power multiplies the indices, so (x³)⁴ is x¹². This is the law most frequently swapped with the multiplication law, because both involve two indices and a single base. The bracket is the reliable signal: brackets multiply the indices, a multiplication sign between separate terms adds them.

The confusion compounds when a coefficient is present. In (2x³)⁴ the four applies to everything inside the bracket, giving 16x¹², and students who apply it only to the x lose the coefficient entirely. Writing the bracket out as four copies once is usually enough to fix the habit permanently.

Anything non-zero raised to the power of zero equals one, and this follows directly from the division law: x³ divided by x³ is x⁰ by the rule, and any non-zero quantity divided by itself is one. Presenting it as an arbitrary fact to memorise is why students doubt it under pressure.

A negative index means a reciprocal, not a negative value. So 2⁻³ is one over 2³, which is one eighth — a positive number. Students who answer negative eight have negated the value instead of inverting it, and this single misreading accounts for a large share of lost marks in the topic.

Combining the two produces expressions like 3x⁻² where only the x carries the negative index, so the result is three over x squared rather than one over three x squared. Bracketing carefully when you rewrite is what keeps the coefficient in the right place.

A fractional index is a root: x to the power of one half is the square root of x, and x to the power of two thirds is the cube root of x squared. This connects two topics that are often taught separately, and recognising the link means surd questions can be answered with index laws instead of a second set of rules.

Senior courses lean on this heavily, particularly in calculus where a term such as one over the square root of x must be rewritten as x to the power of negative one half before it can be differentiated. Students who are fluent with fractional indices find that transition considerably easier than those who are not.

Index laws are unusually vulnerable to being forgotten between the unit and the exam, because they are learned as a list and a list decays quickly. Mixed sets, where consecutive questions require different laws, hold up far better than blocked practice on one law at a time even though they feel harder while you are doing them.

The other habit worth building is checking a doubtful result with small numbers. If you are unsure whether (x³)⁴ is x⁷ or x¹², substitute two for x and evaluate both sides: eight to the fourth is 4096, and two to the twelfth is also 4096, which settles it in seconds without needing to recall the rule.