Practice Maths

L34 — Index Laws and Scientific Notation

Key Terms

Index (exponent)
The power to which a base is raised; in an, n is the index.
Product law
am × an = am+n — add indices when multiplying powers with the same base.
Quotient law
am ÷ an = am−n — subtract indices when dividing powers with the same base.
Negative index
a−n = 1/an — a negative index means take the reciprocal.
Fractional index
am/n = (n√a)m — the denominator is the root, the numerator is the power.
Scientific notation
a × 10n where 1 ≤ a < 10 — used to represent very large or very small numbers concisely.

Index laws summary

LawRuleExample
Productam × an = am+n23 × 24 = 27
Quotientam ÷ an = am−n56 ÷ 52 = 54
Power of a power(am)n = amn(32)4 = 38
Zero indexa0 = 1 (a ≠ 0)70 = 1
Negative indexa−n = 1/an2−3 = 1/8
Fractional indexa1/n = ⁿ√a81/3 = 2
Fractional index (general)am/n = (ⁿ√a)m82/3 = 4

Scientific notation

A number in scientific notation is written as a × 10n where 1 ≤ a < 10 and n is an integer.

NumberScientific notationNotes
6 400 0006.4 × 106Large: positive n
0.000 0323.2 × 10−5Small: negative n
11 × 100n = 0
10⁻³ 0.001 10⁻² 0.01 10⁻¹ 0.1 10⁰ 1 10¹ 10 10² 100 10³ 1000 ×10 ×10 ×10 ×10 ×10 ×10 Powers of 10
Each step right multiplies by 10; each step left divides by 10.
Hot Tip: Index laws only apply when the bases are identical. You cannot add the indices of 23 × 34 — the bases differ. Always check the base before applying a law.

Worked Example 1 — Applying index laws

Simplify: (a) x3 × x5   (b) y8 ÷ y3   (c) (2a2)3

(a) x3 × x5 = x3+5 = x8.

(b) y8 ÷ y3 = y8−3 = y5.

(c) (2a2)3 = 23 × a2×3 = 8a6.

Worked Example 2 — Negative and zero indices

Evaluate: (a) 30   (b) 4−2   (c) (1/2)−3

(a) 30 = 1.

(b) 4−2 = 1/42 = 1/16 = 0.0625.

(c) (1/2)−3 = 23 = 8.

Worked Example 3 — Fractional indices

Evaluate: (a) 271/3   (b) 163/4   (c) 9−1/2

(a) 271/3 = ∛27 = 3.

(b) 163/4 = (∜16)3 = 23 = 8.

(c) 9−1/2 = 1/91/2 = 1/3 ≈ 0.333.

Worked Example 4 — Scientific notation

Write in scientific notation: (a) 45 200 000   (b) 0.000 067. Convert to ordinary: (c) 3.8 × 104.

(a) 45 200 000 = 4.52 × 107.

(b) 0.000 067 = 6.7 × 10−5.

(c) 3.8 × 104 = 38 000.

Worked Example 5 — Operations with scientific notation

Calculate (4 × 105) × (2 × 103) and (6 × 108) ÷ (3 × 105).

(4 × 2) × 105+3 = 8 × 108.

(6 ÷ 3) × 108−5 = 2 × 103.

See Answers ➔

  1. Index laws. Fluency

    • (a) Simplify x4 × x6.
    • (b) Simplify a9 ÷ a4.
    • (c) Simplify (m3)4.
    • (d) Simplify (3b2)3.

  2. Negative and zero indices. Fluency

    • (a) Evaluate 50.
    • (b) Evaluate 2−4.
    • (c) Write 1/x5 as a power of x.
    • (d) Simplify 6x−2 × 2x3.

  3. Fractional indices. Fluency

    • (a) Evaluate 641/2.
    • (b) Evaluate 1251/3.
    • (c) Evaluate 163/4.
    • (d) Evaluate 32−2/5.

  4. Scientific notation. Fluency

    • (a) Write 7 250 000 in scientific notation.
    • (b) Write 0.000 094 in scientific notation.
    • (c) Convert 5.3 × 104 to an ordinary number.
    • (d) Evaluate (3 × 104) × (4 × 103). Express in scientific notation.

  5. Exponential growth pattern. Understanding

    The graph shows the values of 2x for x = −3 to x = 4.

    0 2 4 6 8 16 −3 −2 −1 0 1 2 3 4 x y y = 2ˣ
    • (a) From the graph, state the value of 20.
    • (b) Estimate the value of 23.5 from the graph. (Hint: it lies halfway between x = 3 and x = 4.)
    • (c) Explain why the graph never crosses the x-axis.
    • (d) What is the y-intercept of y = 2x? Explain using index laws.

  6. Mixed simplification. Understanding

    • (a) Simplify (x3y2)2 × x−2.
    • (b) Simplify (4a3) ÷ (2a5).
    • (c) Write √x5 as a fractional index.
    • (d) Simplify (2x2)3 ÷ (4x3).

  7. Scientific notation in context. Understanding

    • (a) The mass of a proton is 1.67 × 10−27 kg. Write this as an ordinary decimal.
    • (b) The distance from Earth to the Sun is approximately 150 000 000 km. Write in scientific notation.
    • (c) A computer processes 2.5 × 109 operations per second. How many operations in 4 seconds? Give your answer in scientific notation.
    • (d) Two bacteria samples have masses 4.2 × 10−6 g and 1.8 × 10−6 g. Find the total mass in scientific notation.

  8. Combining index laws. Understanding

    • (a) Evaluate: 25 × 2−3 ÷ 20.
    • (b) Simplify: (27x3)1/3.
    • (c) Simplify: (x2y−1)3 ÷ (x3y2).
    • (d) Show that am/n = (a1/n)m using the power-of-a-power law.

  9. Doubling and half-life. Problem Solving

    A bacteria colony starts at 500 cells and doubles every hour.

    • (a) Write an expression for the number of cells after t hours.
    • (b) Find the number of cells after 5 hours.
    • (c) Write the number of cells after 5 hours in scientific notation.
    • (d) A radioactive substance has a half-life of 6 hours. Starting at 800 mg, write an expression for the mass remaining after t hours and find the mass after 18 hours.

  10. Index equation solving and estimation. Problem Solving

    • (a) Solve 2x = 64.
    • (b) Solve 3x = 1/27.
    • (c) Find the value of x if (2x)2 = 144.
    • (d) A square has area 200 cm2. Write the side length using a fractional index and evaluate to 2 decimal places.