Year 9 index laws: the mistakes that cost marks in maths tests

Index laws are one of those Year 9 topics that look deceptively manageable in class. Your child nods along, copies the rules into their book, and then loses marks in the test in ways neither of you can quite explain.

The reason is usually not that they never understood it. It's that the rules start to blur together under pressure, especially once negative bases, zero exponents, and fractional indices enter the picture. Here's what's actually going wrong and what to do about it.

Why index laws matter in the Victorian Curriculum

Index laws sit within the Number strand of the Victorian Curriculum 2.0 and underpin a large part of what comes next in algebra, scientific notation, and later in surds and logarithms. Getting shaky foundations here creates compounding problems through Years 10 and beyond. For students aiming at Maths Methods or Specialist, the cost is even higher.

Mistake 1: Adding exponents instead of multiplying the base

When students see something like $2^3 \times 2^4$, the rule is to add the exponents: $2^7$. That part usually sticks. The mistake comes when students apply it to expressions like $2^3 \times 3^4$ and add the exponents anyway, getting $6^7$.

The multiplication law only works when the bases are identical. Mixing bases is one of the most common errors in test conditions, because students are pattern-matching the rule rather than reading the question.

Mistake 2: Mishandling the zero exponent

Most students can tell you that $a^0 = 1$. Ask them why, and they often can't. That gap matters because it means the rule doesn't hold up when the expression gets slightly more complex.

A common mistake is writing $(3x)^0 = 3$ instead of 1, treating the 3 as somehow separate from the exponent. Another is confusing $0^0$ or assuming any base raised to zero must equal zero. When students understand that $a^0 = 1$ comes from the division law ($a^n \div a^n = a^0 = 1$), the rule becomes much harder to misapply.

Mistake 3: Negative bases and even or odd exponents

This is where marks disappear quickly. The expression $(-3)^2$ equals 9, but $-3^2$ equals $-9$. The brackets change everything, and students routinely confuse the two.

The underlying issue is not carelessness. It's that students haven't practised enough examples where the brackets are deliberately absent to make the distinction clear. High-performing students can also get caught here when the expression appears inside a longer simplification and they stop reading carefully.

Mistake 4: The power of a power rule applied incorrectly

The rule $(a^m)^n = a^{mn}$ is reasonably well understood in isolation. It breaks down when students apply it to expressions like $(a^2 b^3)^4$ and only raise the first term, writing $a^8 b^3$ instead of $a^8 b^{12}$.

This is a step-skipping error. Students who work quickly and don't write out each term separately are most vulnerable to it. Extension students should also be comfortable handling negative and fractional exponents inside brackets, where the same error pattern appears in harder forms.

Mistake 5: Negative exponents and what they actually mean

$a^{-2}$ means $\frac{1}{a^2}$, not $-a^2$. This is consistently one of the most misunderstood rules in the topic.

Students who have memorised the result without understanding its origin often write the sign in the wrong place, especially in multi-term expressions. Ask your child to explain in words what $x^{-3}$ means. If they can't, that's the gap to fix before the test.

Mistake 6: Fractional indices

Fractional indices are where students who have managed the rest of the topic tend to drop marks. $a^{\frac{1}{2}} = \sqrt{a}$ and $a^{\frac{m}{n}} = \sqrt[n]{a^m}$ are rules that require both index law knowledge and comfort with surds.

The most common mistake is reversing the numerator and denominator: treating $a^{\frac{2}{3}}$ as $\sqrt{a^3}$ rather than $(\sqrt[3]{a})^2$ or equivalently $\sqrt[3]{a^2}$. For students sitting extension assessments or preparing for Year 10 maths, this is worth spending real time on.

What actually fixes these errors

Reading through the rules again rarely helps, because students can usually recite the rules. What helps is working through questions they get wrong and being shown the exact step where the logic broke down.

A common pattern is students who revise by reading worked examples, feel confident, and then make the same errors in the test. Confidence built on recognition ("I remember seeing this") is not the same as being able to do it from scratch. The fix is generating fresh questions on the exact concept that's slipping and repeating until the method is automatic.

For students who are genuinely strong at this topic, the extension move is to mix multiple index laws in the one expression and introduce algebraic bases with multiple variables. That's where the interesting problems live, and where a student can genuinely separate themselves from the cohort.

Turn the questions they get wrong into practice

Got some of these wrong? Snap them in Tuterly and get a fresh version to practise until they're mastered. It works alongside a tutor or on its own.