Practice Maths

Problem Solving with Indices

Key Terms

Scientific notation
Writing a number as a × 10n where 1 ≤ a < 10; used for very large or very small numbers.
Standard form
The ordinary way of writing a number without powers of 10; e.g. 4 500 000 rather than 4.5 × 106.
Hot Tip Moving the decimal point to the right increases the power of 10 (large numbers). Moving it to the left decreases the power (small numbers, negative indices).

Worked Example

Convert 67 400 to scientific notation.

Move the decimal point left until you have a number between 1 and 10:
6.74 × 104  (moved 4 places left)

Convert 2.3 × 10−4 to standard form.

Negative index means small number — move decimal 4 places left:
0.00023

Why Scientific Notation Exists

The mass of an electron is 0.000 000 000 000 000 000 000 000 000 000 911 kg. The distance from Earth to the Andromeda Galaxy is 24 000 000 000 000 000 000 000 metres. Writing these in full is impractical and error-prone (it's easy to lose a zero). Scientific notation solves this by expressing any number as a × 10n, where a is between 1 and 10 (including 1, not including 10), and n is an integer. The electron mass becomes 9.11 × 10−31 kg — compact and clear.

The rule for a is critical: 1 ≤ a < 10. So 45 × 103 is NOT in correct scientific notation (45 ≥ 10), but 4.5 × 104 is. Similarly, 0.6 × 105 is not correct (0.6 < 1), but 6 × 104 is.

Converting to Scientific Notation

To convert 8 600 000 to scientific notation: place the decimal point after the first significant digit to get 8.6, then count how many places you moved the decimal left (6 places). Result: 8.6 × 106. For small numbers like 0.000047: move the decimal right to get 4.7, counting the moves (5 places). Because you moved right, the index is negative: 4.7 × 10−5.

A useful check: large numbers have positive indices, small numbers (between 0 and 1) have negative indices. 4.5 × 106 = 4 500 000 (big). 4.5 × 10−6 = 0.0000045 (tiny).

Key Tip: Moving the decimal left (to make a smaller number from the digits) gives a positive index — the original number was large. Moving the decimal right gives a negative index — the original was small. Large number = positive power; tiny number = negative power.

Converting from Scientific Notation

To expand 3.72 × 104: multiply by 104, which means move the decimal 4 places right: 37 200. For 6.1 × 10−3: move decimal 3 places left: 0.0061. Count carefully — zeros must be inserted as placeholders.

Comparing numbers in scientific notation: first compare the powers of 10. 3.5 × 108 versus 9.1 × 107. The first has power 8, the second power 7, so 3.5 × 108 is larger (even though 9.1 > 3.5, the power of 10 dominates).

Key Tip: When comparing scientific notation values, compare powers of 10 first. A higher power of 10 means a larger number, regardless of the leading digit. Only compare the leading digits (a values) when the powers are equal.

Calculating with Scientific Notation

To multiply: (3 × 104) × (2 × 103) = (3 × 2) × (104 × 103) = 6 × 107. Multiply the a values together, and add the indices. If the result for a falls outside [1, 10), adjust: (5 × 104) × (3 × 103) = 15 × 107 = 1.5 × 108 (because 15 = 1.5 × 101). Scientific notation is the language of science — mastering it means you can work with any scale of measurement confidently.

Mastery Practice

  1. Convert each number to scientific notation. Fluency

    1. 3 400
    2. 72 000
    3. 850 000
    4. 4 200 000
    5. 0.06
    6. 0.00085
    7. 0.000047
    8. 56 000 000

  2. Convert each number from scientific notation to standard form. Fluency

    1. 2.5 × 103
    2. 8.1 × 105
    3. 3.72 × 107
    4. 6.04 × 104
    5. 1.8 × 10−2
    6. 4.5 × 10−4
    7. 9.9 × 10−6
    8. 7.3 × 109

  3. Arrange each set in ascending order (smallest to largest). Fluency

    1. 3.2 × 104,   8.1 × 103,   1.5 × 105
    2. 6.7 × 10−2,   4.1 × 10−3,   9.3 × 10−4
    3. 2.0 × 106,   7.5 × 105,   3.1 × 106,   9.9 × 104
    4. 5.4 × 10−5,   1.2 × 10−4,   8.8 × 10−5
    5. 1.0 × 103,   9.9 × 102,   1.1 × 103
    6. 4.0 × 108,   3.9 × 109,   5.2 × 107,   6.1 × 108

  4. Calculate each expression. Give your answer in scientific notation. Understanding

    1. (3 × 104) × (2 × 103)
    2. (8 × 106) ÷ (4 × 102)
    3. (5 × 10−3) × (3 × 105)
    4. (9 × 108) ÷ (3 × 104)
    5. (4 × 103)2
    6. (6 × 10−2) × (5 × 104)
    7. (2.4 × 105) ÷ (1.2 × 102)
    8. (7 × 104) × (3 × 10−6)

  5. Mixed index law problems — simplify fully using all index laws as needed. Understanding

    1. (32)3 × 34 ÷ 38
    2. 27 × 2−3
    3. 54 ÷ 56 (write as a negative index, then as a fraction)
    4. (103)4 ÷ (105)2
    5. 4−2 × 43
    6. (63)2 ÷ (64)2
    7. 75 × 7−2 ÷ 71
    8. (2−1)3

  6. Apply index notation and scientific notation to real-world contexts. Problem Solving

    1. The population of Australia is approximately 26 million. Write this in scientific notation.
    2. The distance from Earth to the Sun is about 1.496 × 108 km. Write this as a standard number.
    3. A hard drive has a capacity of 2 terabytes. If 1 terabyte = 1012 bytes, write the drive capacity in scientific notation.
    4. Light travels at approximately 3 × 105 km per second. How far does light travel in 60 seconds? Give your answer in scientific notation.
    5. A bank account earns interest. After n years the balance is $1000 × (1.05)n. Explain whether this is in index notation, and calculate the balance after 2 years (you may use a calculator for the final value).
    6. Two atoms have masses of 3.2 × 10−24 g and 8.0 × 10−23 g. Which is heavier, and by how many times?

  7. State whether each expression is in correct scientific notation. If not, rewrite it correctly. Fluency

    1. 3.5 × 104
    2. 15 × 103
    3. 0.7 × 106
    4. 9.1 × 10−2
    5. 10 × 105
    6. 1.0 × 100
    7. 6.28 × 10−7
    8. 0.04 × 108

  8. Simplify using index laws. Express answers in index notation. Fluency

    1. (32)4 × 32
    2. 28 ÷ (23)2
    3. (53)2 × 50 ÷ 54
    4. [(42)3 × 41] ÷ 45
    5. 37 × 3−3
    6. 104 ÷ 107 (write as a negative index)

  9. Solve each problem using scientific notation. Show all working. Understanding

    1. A factory produces 4.8 × 105 items per day. How many items are produced in 30 days? Give your answer in scientific notation.
    2. A data file is 6.4 × 109 bytes. If there are 1.6 × 103 sections in the file, how large is each section? Give your answer in scientific notation.
    3. The radius of the Earth is approximately 6.4 × 103 km. Use the formula C = 2πr to estimate the circumference (use π ≈ 3.14). Give your answer in scientific notation.
    4. A molecule has a diameter of 5 × 10−9 m. How many molecules placed end to end would fit across a 1 mm gap? (Hint: 1 mm = 10−3 m)

  10. Investigation: Powers and scale. Problem Solving

    1. The width of a human hair is about 7 × 10−5 m. The width of a DNA strand is about 2 × 10−9 m. How many times wider is a human hair than a DNA strand? Show your working using index laws.
    2. Planet X has a mass of 4.5 × 1024 kg and Planet Y has a mass of 9.0 × 1022 kg. How many times more massive is Planet X? Write your answer in scientific notation.
    3. A scientist claims: “(2.5 × 103)2 = 6.25 × 106.” Verify whether this claim is correct, showing full working.
    4. Research task: Look up one real-world measurement that uses scientific notation (e.g. a distance in space, a particle mass, a national debt). Write it down and explain what the number means in plain language.
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