Skip to content
SuccessHomeschool
Menu
Log inStart free

Year 9 · Mathematics · Number

Index laws

Use index notation and the index laws for multiplying, dividing and raising powers, including the zero index, with numbers and pronumerals.

Awaiting educator review. This material was drafted by our content team with AI assistance and is waiting for review by a qualified educator. Please check it suits your child before using it. Created 10 October 2026.

These lessons were written around Australian Curriculum strands and matched to New Zealand year levels by age (NZ Year 1 is the first year of school). They are not yet mapped to The New Zealand Curriculum, so check them against your own learning programme.

Learning goals

Learners will

  • Write repeated multiplication using index notation.
  • Apply the multiplication, division and power-of-a-power index laws.
  • Understand why any non-zero number to the power of zero equals 1.

Success looks like

  • I can identify the base and the index in an expression like 5^3.
  • I can simplify expressions such as 3x^4 × 5x^2 and 20a^7 ÷ 4a^3.
  • I can simplify a power of a power, such as (2m^3)^4.
  • I can evaluate expressions that include a zero index.

The big idea

Index notation is a shorthand for repeated multiplication. Instead of writing 2 × 2 × 2 × 2 × 2, we write 2^5 (read as "2 to the power of 5"). The 2 is the base and the 5 is the index. On this site we write powers with the ^ symbol, so x^3 means x × x × x.

Multiplying with the same base: add the indices. For example, 2^3 × 2^4 means (2 × 2 × 2) × (2 × 2 × 2 × 2), which is seven 2s multiplied together, so 2^3 × 2^4 = 2^7. In general, a^m × a^n = a^(m + n).

Dividing with the same base: subtract the indices. For example, 5^6 ÷ 5^2: six 5s on top and two 5s underneath; two pairs cancel, leaving four 5s, so the answer is 5^4. In general, a^m ÷ a^n = a^(m − n).

Power of a power: multiply the indices. (x^2)^3 means x^2 × x^2 × x^2, which is x^6. In general, (a^m)^n = a^(m × n). When a bracket contains a product, every factor inside is raised to the power, so (3a)^2 = 3^2 × a^2 = 9a^2.

The zero index: look at the pattern 2^3 = 8, 2^2 = 4, 2^1 = 2. Each time the index drops by 1 we divide by 2, so the next value is 2^0 = 1. The division law gives the same result: a^4 ÷ a^4 = a^0, and anything divided by itself is 1. So a^0 = 1 for any non-zero a.

When terms have coefficients, deal with the numbers and the pronumerals separately. Multiply (or divide) the coefficients as normal numbers, then use the index laws for the pronumerals.

  • The index laws only work when the bases are the same. 2^3 × 3^2 cannot be combined into a single power.

Worked examples

Multiplying terms with coefficients

Simplify 3x^4 × 5x^2.

  1. Multiply the coefficients: 3 × 5 = 15.
  2. The pronumeral parts have the same base, so add the indices: x^4 × x^2 = x^(4 + 2) = x^6.
  3. Combine the two parts.

Answer: 15x^6

Dividing terms with coefficients

Simplify 20a^7 ÷ 4a^3.

  1. Divide the coefficients: 20 ÷ 4 = 5.
  2. Subtract the indices for the same base: a^7 ÷ a^3 = a^(7 − 3) = a^4.
  3. Combine the two parts.

Answer: 5a^4

Raising a product to a power

Simplify (2m^3)^4.

  1. Every factor inside the bracket is raised to the power of 4.
  2. The number part: 2^4 = 2 × 2 × 2 × 2 = 16.
  3. The pronumeral part: (m^3)^4 = m^(3 × 4) = m^12.
  4. Combine the two parts.

Answer: 16m^12

Practice check

Have a go, then check your answers. Each answer comes with an explanation.

  1. 1.Simplify 2^3 × 2^4.
  2. 2.Simplify y^9 ÷ y^3.
  3. 3.Simplify (x^2)^5.
  4. 4.What is the value of 7^0?
  5. 5.Simplify (3a)^2.
  6. 6.Evaluate 5^0 + 2^3.
  7. 7.Simplify 6x^5 × 2x^3. Write powers using ^, for example x^2.
  8. 8.Write 3^4 as an ordinary number.
Answer guide for parents

  1. Simplify 2^3 × 2^4.

    2^7 — Same base, so add the indices: 3 + 4 = 7. The answer is 2^7.

  2. Simplify y^9 ÷ y^3.

    y^6 — Same base, so subtract the indices: 9 − 3 = 6. The answer is y^6.

  3. Simplify (x^2)^5.

    x^10 — For a power of a power, multiply the indices: 2 × 5 = 10. The answer is x^10.

  4. What is the value of 7^0?

    1 — Any non-zero number raised to the power of 0 equals 1.

  5. Simplify (3a)^2.

    9a^2 — Both factors inside the bracket are squared: 3^2 × a^2 = 9a^2.

  6. Evaluate 5^0 + 2^3.

    9 — 5^0 = 1 and 2^3 = 8, so the total is 1 + 8 = 9.

  7. Simplify 6x^5 × 2x^3. Write powers using ^, for example x^2.

    12x^8 — Multiply the coefficients (6 × 2 = 12) and add the indices (5 + 3 = 8) to get 12x^8.

  8. Write 3^4 as an ordinary number.

    81 — 3^4 = 3 × 3 × 3 × 3 = 9 × 9 = 81.

Watch out for

  • Multiplying the base by the index, such as thinking 2^3 = 6. Write the repeated multiplication out in full (2 × 2 × 2 = 8) until the meaning is secure.
  • Multiplying the indices when multiplying terms, such as writing x^4 × x^2 = x^8. Expand both terms to count the factors and see that there are six xs, not eight.
  • Thinking that a^0 = 0. Use the halving pattern (8, 4, 2, 1) to show why the answer is 1.
  • Forgetting to apply the power to the coefficient, such as writing (3a)^2 = 3a^2. Expand it as 3a × 3a to see that the 3 is squared too.

Tips for parents

  • When your learner is unsure, ask them to write the expression out as repeated multiplication. This turns the rule back into something they can count.
  • Encourage them to check numerical answers with a calculator using the power key, but to work the index laws out by hand first.
  • Ask your learner to explain each law in their own words; explaining is a strong sign of real understanding.

Go further

  • Fold a sheet of paper in half repeatedly. Record the number of layers after each fold as a power of 2 and predict how many layers there would be after 10 folds (2^10 = 1024).
  • Investigate what a negative index might mean by continuing the halving pattern below 2^0: 2^−1, 2^−2 and so on.
  • Find out how computer storage sizes relate to powers of 2, and write a short explanation for a younger sibling.