VCE Vault

Year 9 Mathematics · Number and Algebra · 18m read

Index Laws and Scientific Notation

Ever wondered how scientists write the distance to the Sun without filling a page with zeros? Powers (also called indices) let you compress enormous or tiny numbers into scientific notation—a clever shorthand that makes calculations way easier. You'll master the rules for multiplying and dividing powers, then use them to solve real problems in physics and beyond.

Learning objective: Apply the index laws and express numbers in scientific notation.

By the end of this lesson you can:

  • I apply all index laws confidently.
  • I convert to and from scientific notation.
  • I calculate with scientific notation.

Index laws are shortcuts on a long road. Instead of multiplying 3 forty times, write 3⁴⁰. Index laws let you combine these shortcuts without ever writing the long version.

Index Laws + Scientific Notation

Index laws let you work with powers that share the same base. Scientific notation tames very large and very small numbers by writing them as $a \times 10^n$.

### The Seven Laws

| Law | Rule | Example |
|-----|------|---------|
| Multiplication | $a^m \times a^n = a^{m+n}$ | $x^3 \times x^4 = x^7$ |
| Division | $a^m \div a^n = a^{m-n}$ | $x^5 \div x^2 = x^3$ |
| Power of a power | $(a^m)^n = a^{mn}$ | $(x^2)^3 = x^6$ |
| Zero index | $a^0 = 1$ | $7^0 = 1$ |
| Negative index | $a^{-n} = \dfrac{1}{a^n}$ | $2^{-3} = \dfrac{1}{8}$ |
| Power of a product | $(ab)^n = a^n b^n$ | $(3x)^2 = 9x^2$ |
| Power of a quotient | $\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$ | $\left(\dfrac{x}{2}\right)^3 = \dfrac{x^3}{8}$ |

### Why indices matter

Index notation is the language of science, technology and finance. From bacteria doubling every hour ($N = N_0 \times 2^t$) to compound interest ($A = P(1+r)^n$), exponents describe multiplicative growth.

### Scientific Notation

Scientific notation expresses any number as $a \times 10^n$ where $1 \leq a < 10$.

#### Large Numbers (positive exponent)

  • $34\,000 = 3.4 \times 10^4$ (4 places left)
  • $5\,600\,000 = 5.6 \times 10^6$ (6 places left)
  • The diameter of Earth $\approx 1.27 \times 10^7$ m

#### Small Numbers (negative exponent)

  • $0.0072 = 7.2 \times 10^{-3}$ (3 places right)
  • $0.000\,000\,001$ m $= 1 \times 10^{-9}$ m (1 nanometre!)
  • A hydrogen atom is about $5.3 \times 10^{-11}$ m across

### Calculating with Scientific Notation

#### Multiply

$(3 \times 10^4) \times (2 \times 10^3) = (3 \times 2) \times 10^{4+3} = 6 \times 10^7$

#### Divide

$(8 \times 10^5) \div (4 \times 10^2) = (8 \div 4) \times 10^{5-2} = 2 \times 10^3$

> Important: If the digit part falls outside $[1, 10)$, adjust the exponent — e.g. $24 \times 10^3 = 2.4 \times 10^4$.

### Australian Context

Australia has about $2.7 \times 10^7$ people. The distance from Melbourne to Sydney is roughly $8.8 \times 10^2$ km. The national debt is often expressed in scientific notation to avoid confusing zeroes.

Key Index Law Formulas

  • Multiply same base: a^m \times a^n = a^{m+n}
  • Divide same base: a^m \div a^n = a^{m-n}
  • Power of power: (a^m)^n = a^{mn}
  • Zero index: a^0 = 1 \quad (a \neq 0)
  • Negative index: a^{-n} = \dfrac{1}{a^n}
  • Scientific notation form: a \times 10^n, \quad 1 \leq a < 10 ($n$ is an integer)

Remember

  • Different bases: Index laws DON'T apply directly — $2^3 \times 3^3 \neq 6^6$.
  • 1 ≤ a < 10: Coefficient must be in this range for scientific notation.
  • Estimation check: Always sanity-check the exponent by counting decimal places.

Worked example (Medium)

Simplify: $(2x^3)^2 \times 3x^4 \div 4x^2$

  1. Square the bracket. $(2x^3)^2 = 2^2 \times (x^3)^2 = 4x^6$
  2. Multiply. $4x^6 \times 3x^4 = 12x^{10}$
  3. Divide. $12x^{10} \div 4x^2 = 3x^8$

Answer: $3x^8$

Worked example (Hard)

Simplify fully: $\dfrac{6a^{-2}b^3 \times (2ab)^2}{3a^2b^{-1}}$

  1. Expand bracket. $(2ab)^2 = 4a^2b^2$
  2. Numerator. $6a^{-2}b^3 \times 4a^2b^2 = 24a^{-2+2}b^{3+2} = 24a^0b^5 = 24b^5$
  3. Divide. $\dfrac{24b^5}{3a^2b^{-1}} = 8a^{-2}b^{5-(-1)} = 8a^{-2}b^6$
  4. Positive indices. $8a^{-2}b^6 = \dfrac{8b^6}{a^2}$

Answer: $\dfrac{8b^6}{a^2}$

Common mistakes

  • Writing $x^3 \times x^4 = x^{12}$ (multiplying the indices) The multiplication law says ADD indices when multiplying same base. Multiplying indices only applies to $(x^3)^4$. Fix: $x^3 \times x^4 = x^{3+4} = x^7$. Reserve multiplication of indices for a power raised to a power.
  • Forgetting to adjust the coefficient when converting to scientific notation — leaving $24 \times 10^3$ Scientific notation requires the digit part to be between 1 and 10. $24$ is outside this range. Fix: Move the decimal one place: $24 \times 10^3 = 2.4 \times 10^4$.
  • Computing $3^{-2} = -9$ (applying a minus sign to the value) A negative index means RECIPROCAL, not a negative number. Fix: $3^{-2} = \dfrac{1}{3^2} = \dfrac{1}{9}$, which is positive.

Practice questions

  1. $(x^5)^3 = ?$

    1. $x^8$
    2. $x^{15}$
    3. $x^2$
    4. $x^{25}$

    (1 marks)

  2. Write 0.00045 in scientific notation.

    (2 marks)

  3. Simplify $\dfrac{x^8}{x^3}$ and evaluate when $x = 2$.

    (2 marks)

  4. Calculate $(4.5 \times 10^6) \times (2 \times 10^{-4})$. Express in scientific notation.

    (3 marks)

  5. Simplify: $\dfrac{(3x^2)^3 \times x^{-1}}{9x^4}$

    (3 marks)

Going Deeper — Fractional Indices

Indices do not have to be integers. Fractional indices connect back to surds (roots):

$a^{1/2} = \sqrt{a}$
$a^{1/3} = \sqrt[3]{a}$
$a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$

Examples:
• $8^{1/3} = \sqrt[3]{8} = 2$
• $16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8$
• $25^{-1/2} = \dfrac{1}{\sqrt{25}} = \dfrac{1}{5}$

This is a preview of Year 10 content — understanding that roots and powers are unified by the index laws is a key insight.

In physics, the inverse-square law for gravity uses $F \propto r^{-2}$. In statistics, the standard deviation of a sample mean involves $n^{-1/2}$. Fractional indices are everywhere once you look for them.

Summary

  • The seven index laws all require the SAME BASE to apply.
  • $a^0 = 1$ for any non-zero $a$; $a^{-n} = 1/a^n$ (reciprocal, not negative value).
  • Scientific notation: $a \times 10^n$ where $1 \leq a < 10$.
  • Multiply in SN: multiply digits, ADD exponents.
  • Divide in SN: divide digits, SUBTRACT exponents.
  • Always adjust if the digit part falls outside $[1, 10)$ after a calculation.

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