Index laws are seven rules for simplifying expressions built from powers of the same base. Australian students first meet them in Year 8, where the curriculum introduces positive-integer and zero indices, and use them through senior maths and first-year university algebra.
The rules never change. Index, power, and exponent all name the same thing: the small raised number that counts how many times a base multiplies itself. In 2^5, the base is 2 and the index is 5.
What are index laws?
An index law is a shortcut. Instead of expanding 3^4 × 3^2 into (3×3×3×3)×(3×3), you keep the base and add the indices to reach 3^6.
The catch is the base. The laws only work when the base is identical.
That single condition trips up more students than any individual rule. 2^3 × 3^2 cannot be combined, because 2 and 3 are different bases.
Across exam boards the rules are stable. The same seven laws appear in the Australian Curriculum descriptor VCMNA272, in IGCSE and A-Level specifications, and in the algebra section of the digital SAT. Learn them once and they transfer everywhere.
What are the seven index laws?
Each law below assumes the same base, and a base that is not zero. The first three handle multiplication, division, and nested powers. The last four cover brackets, the zero case, and negatives.
Read the table as patterns, not facts to memorise cold. Multiplication adds the indices, division subtracts them, a power of a power multiplies them.
The structure repeats, which is why spaced repetition works so well here. AskSia's Flashcards build a deck from your own notes and schedule reviews with FSRS, so the seven rules are locked in well before an exam, not the night before.
How do you apply the laws?
Most problems chain two or three laws together. Work base by base, simplify each group, then collect what is left.
Take (x^5 × x^2) ÷ x^3. The product law handles the top: x^5 × x^2 = x^7. The quotient law finishes it: x^7 ÷ x^3 = x^4. One expression, two laws.
Nested powers need the third law first. In (2^3)^4, multiply the indices to get 2^12, or 4096. Add them by mistake and you land on 2^7, which is 128 — a gap of nearly 4000 from one wrong operation.
Coefficients travel on a separate track. For 6a^4 × 2a^3, multiply the numbers (6 × 2 = 12) and add the indices (a^(4+3) = a^7), giving 12a^7. The number and the power are never mixed.
This is where practice volume beats theory. AskSia's Mock Exam mode generates index questions in the format of your actual paper and grades each step with a rationale, so a wrong move shows up before the exam rather than after.
What do negative and fractional indices mean?
These two cause the most confusion. They break the intuition that an index simply counts repeated multiplication.
A negative index means reciprocal, not a negative answer. 2^(−3) = 1 ÷ 2^3 = 1/8, a positive value. The minus sign moves the power to the denominator. 5^(−1) is just 1/5.
A fractional index means a root. The denominator is the root, the numerator is the power. So 8^(2/3) is the cube root of 8, squared: (∛8)^2 = 2^2 = 4. And 9^(1/2) is √9 = 3.
The zero case falls out of the quotient law. Any nonzero base to the power 0 equals 1, because a^n ÷ a^n = a^0, and any number divided by itself is 1. So 7^0, 100^0, and (−3)^0 all equal 1.
When one step refuses to click, the AI tutor re-explains it three ways until one lands — a numeric example, an algebraic derivation, then a visual. For fractional indices, the surd-to-power visual is usually the one that fixes the idea.
Where do index laws appear?
Index laws are not a Year 8 topic you leave behind. They sit underneath scientific notation, compound interest, surds, logarithms, and the calculus power rule.
Scientific notation is index laws in disguise. Multiplying (3 × 10^8) by (2 × 10^5) applies the product law to the powers of 10, giving 6 × 10^13. Physics and chemistry lean on this constantly.
Compound interest is exponential growth. A balance multiplied by 1.05 each year for 10 years is balance × 1.05^10, roughly a 63% rise. Every superannuation projection runs on the same arithmetic.
Logarithms are the inverse of indices, and surds are fractional indices in different clothing. The power rule in calculus brings the index down and subtracts one. That is why Calculus 2 at UniMelb (MAST10006) and first-year units like Monash's Foundation Mathematics assume index fluency from day one.
AskSia's Concept Map lays out these links as a tree, so index laws visibly feed surds, then logs, then the calculus power rule, instead of sitting as separate topics you relearn each year.
What mistakes do students make?
Most index errors are not careless slips. They come from applying a real rule in the wrong place, usually around bases, signs, or the zero index.
The table pairs the four most common wrong moves with the fix. Almost every fix is the same: re-check the base.
Notice the pattern. The index arithmetic itself is rarely the problem. Slow down on the base first, and most of these errors disappear.
Frequently Asked Questions
What are the laws of indices?
The laws of indices, also called index laws or exponent rules, are the seven rules for combining powers that share a base. They cover multiplication (add the indices), division (subtract them), a power raised to a power (multiply them), brackets, the zero index, and negative indices. The Australian Curriculum introduces them in Year 8 under descriptor VCMNA272, and the same seven appear in IGCSE, A-Level, and the digital SAT algebra section. They only apply when the base is identical: 2^3 × 2^4 = 2^7 works, but 2^3 × 3^2 does not simplify because the bases differ. Once learned, they carry straight into surds, logarithms, and calculus. For a one-page reference while you practise, the MAT9004 cheatsheet lays the rules out with worked examples.
What are the 7 index laws?
The seven are: the product law (a^m × a^n = a^(m+n)), the quotient law (a^m ÷ a^n = a^(m−n)), the power of a power ((a^m)^n = a^(mn)), the power of a product ((ab)^n = a^n b^n), the power of a quotient ((a/b)^n = a^n / b^n), the zero index (a^0 = 1), and the negative index (a^(−n) = 1/a^n). The fractional index, a^(m/n) = the nth root of a^m, is often listed as an eighth rule but follows from the same logic. All assume the base is not zero. To drill them in exam format with step-by-step feedback, run a set through AskSia's Mock Exam mode.
What does a negative index mean?
A negative index means reciprocal, not a negative number. The rule is a^(−n) = 1/a^n, so the power moves to the denominator and the value stays positive. For example, 2^(−3) = 1/2^3 = 1/8, and 5^(−1) = 1/5. A common error is writing 2^(−2) = −4; the correct answer is 1/4. Negative indices appear constantly in scientific notation, where a number like 4 × 10^(−6) means 0.000004, and in unit conversions in physics. The fastest way to stop confusing the sign is to expand one example by hand, then rely on the rule. If a single step keeps tripping you up, AskSia's AI tutor will re-derive it two or three different ways until the reciprocal idea sticks.
What is a fractional index equal to?
A fractional index is a root. In a^(m/n), the denominator n is the root and the numerator m is the power. So 9^(1/2) = √9 = 3, and 8^(2/3) means the cube root of 8, then squared: (∛8)^2 = 2^2 = 4. Order does not matter; you can square first then take the root and reach the same answer. This is the bridge between surds and indices, which is why senior maths courses ask you to switch between √x and x^(1/2) freely. Fractional indices are heavily tested in HSC, VCE, and SAT algebra. Build a few flashcards pairing each surd with its index form, and AskSia's Flashcards with FSRS will resurface them on the schedule that matches your exam date.
What is the zero index rule?
The zero index rule states that any nonzero base raised to the power 0 equals 1. So 7^0 = 1, 100^0 = 1, and (−3)^0 = 1. It follows directly from the quotient law: a^n ÷ a^n = a^(n−n) = a^0, and since any nonzero number divided by itself is 1, a^0 must equal 1. The one exception is 0^0, which is left undefined in most syllabuses. A frequent mistake is writing x^0 = 0 instead of 1, which then breaks every later step. For the full set of foundation rules tied to a real first-year unit, see the Monash Foundation Mathematics course page.
How do you simplify expressions with indices?
Work one base at a time. Group the powers of each base, apply the product and quotient laws within each group, then handle any brackets and negatives last. For (x^5 × x^2) ÷ x^3, add 5 and 2 to get x^7, then subtract 3 to get x^4. Keep coefficients separate from indices: in 6a^4 × 2a^3, multiply 6 × 2 = 12 and add the indices to get 12a^7. Leave answers with positive indices unless told otherwise, converting any negative power using a^(−n) = 1/a^n. The reliable path is volume: work 20 to 30 mixed questions until the base-by-base habit is automatic. AskSia's Mock Exam mode can generate that set in your paper's format and flag the exact step where a simplification went wrong.
When do index laws not apply?
The laws only hold for the same base. They say nothing about adding or subtracting powers, which is a separate operation. 2^3 + 2^2 is not 2^5; it is 8 + 4 = 12.
They also will not merge different bases into one. 2^3 × 5^2 stays as it is, or becomes 8 × 25 = 200 by direct calculation, never 10 to some power. The same-base condition is the boundary of the entire system.
