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Powers Roots And Standard Form

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Notes de leçon

Powers & Roots

  • Powers (indices) show repeated multiplication: e.g., 6³ = 6 × 6 × 6.
  • Any non-zero number to the power of 0 equals 1: 3⁰ = 1.
  • Any number to the power of 1 equals itself: 3¹ = 3.
  • Square roots reverse squaring; every positive number has two square roots (positive and negative). Notation √ refers to the positive root.
  • Cube roots reverse cubing; each number has exactly one real cube root (∛).
  • nth roots: if n is even, positive numbers have two real nth roots; negative numbers have none. If n is odd, every number has one real nth root.
  • Reciprocal of a number is 1 divided by that number; written as index -1 (e.g., 5⁻¹ = 1/5).

Laws of Indices

  • Multiplication: aᵐ × aⁿ = aᵐ⁺ⁿ (add powers).
  • Division: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (subtract powers).
  • Power of a power: (aᵐ)ⁿ = aᵐⁿ (multiply powers).
  • Negative power: a⁻ⁿ = 1/aⁿ (reciprocal).
  • Fractional power: a1/n = ⁿ√a; am/n = (ⁿ√a)ᵐ = ⁿ√(aᵐ).
  • Zero power: a⁰ = 1 (for a ≠ 0).
  • Product and quotient rules: (ab)ⁿ = aⁿbⁿ; (a/b)ⁿ = aⁿ/bⁿ.

Converting to Standard Form

  • Standard form: A × 10ⁿ, where 1 ≤ A < 10 and n is an integer.
  • For large numbers (≥10), n is positive: count how many places the decimal moves left. Example: 32400 = 3.24 × 10⁴.
  • For small numbers (0 < number < 1), n is negative: count how many places the decimal moves right. Example: 0.0000324 = 3.24 × 10⁻⁵.
  • To convert from standard form to ordinary number, move the decimal point n places (right if n>0, left if n<0).

Converting a large number to standard form

Converting a large number to standard form

Operations with Standard Form

  • Multiplication: multiply the A parts, add the powers of 10, then adjust to standard form. Example: (3×10²)×(4×10⁵)=12×10⁷=1.2×10⁸.
  • Division: divide the A parts, subtract the powers of 10, then adjust. Example: (2×10⁻⁵)÷(8×10⁻³)=0.25×10⁻²=2.5×10⁻³.
  • Addition/Subtraction (Method 1): convert both to ordinary numbers, add/subtract, then convert back to standard form.
  • Addition/Subtraction (Method 2): rewrite numbers to have the same power of 10, add/subtract the A parts, then adjust. Efficient for large or negative exponents.
  • On a calculator, use brackets and the ×10ˣ button to enter standard form numbers.

Common Mistakes & Tips

  • Index laws only work with same base; e.g., 2³×5² cannot be simplified directly.
  • A negative power means reciprocal, not a negative number: 2⁻⁴ = 1/16, not -16.
  • When taking even roots of positive numbers, remember there are two roots (e.g., √25 = ±5), but the radical sign denotes the principal (positive) root.
  • In standard form, A must be ≥1 and <10; if not, adjust by moving the decimal and changing the exponent accordingly.

Diapos

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  1. 1.Write 510 100 000 in standard form.

    Easy
    • A5.101 × 108
    • B5.101 × 109
    • C51.01 × 107
    • D5.101 × 107

  2. 2.Write 0.00527 in standard form.

    Easy
    • A5.27 × 10-3
    • B5.27 × 10-4
    • C5.27 × 103
    • D52.7 × 10-4

  3. 3.Write 3.4 × 10-1 as an ordinary number.

    Easy
    • A0.34
    • B34
    • C3.4
    • D0.034

  4. 4.Which of these numbers is the largest? 3.4 × 10-1, 1.36 × 106, 7.9 × 100, 2.4 × 105, 5.21 × 10-3, 4.3 × 10-2

    Easy
    • A1.36 × 106
    • B2.4 × 105
    • C7.9 × 100
    • D3.4 × 10-1

  5. 5.Find the value of (2/5)2.

    Easy
    • A4/25
    • B2/25
    • C4/5
    • D2/5

  6. 6.Work out (5.2 × 107) + (5.2 × 106). Give your answer in standard form.

    Medium
    • A5.72 × 107
    • B5.2 × 107
    • C5.72 × 106
    • D1.04 × 108

  7. 7.Work out (9 × 10-4) × (3 × 107). Give your answer in standard form.

    Medium
    • A2.7 × 104
    • B2.7 × 103
    • C2.7 × 105
    • D27 × 103

  8. 8.Patrick says that 641/4 = 16 because 1/4 of 64 is 16. What is wrong with his reasoning?

    Medium
    • A641/4 means the fourth root of 64, not 1/4 of 64
    • B641/4 = 64 ÷ 4 = 16
    • C641/4 = 64 × 1/4 = 16
    • DNothing, Patrick is correct

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