Flashcards · J560 · Topic 3

Indices and surds flashcards

15 cards on Topic 3, subtopics 3.01 to 3.03. Pick a subtopic or study them all.

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Laws of indices (same base): aᵐ × aⁿ = aᵐ⁺ⁿ aᵐ ÷ aⁿ = aᵐ⁻ⁿ (aᵐ)ⁿ = aᵐⁿ a⁰ = 1 for any non-zero a. Different bases do not combine: 2³ × 3² = 8 × 9 = 72.
Negative indices: 3⁻² and (2/3)⁻³: A negative index means the reciprocal, not a negative number. 3⁻² = 1/9 (2/3)⁻³ = (3/2)³ = 27/8
Simplify (2a³)⁴: The power applies to the number and every letter: 2⁴ × a¹² = 16a¹².
Fractional indices: 32^(2/5) and 8^(−1/3)Higher: The denominator is the root, the numerator the power; a negative sign means reciprocal. 32^(2/5) = (fifth root of 32)² = 2² = 4 8^(−1/3) = 1/∛8 = 1/2
Estimate √200 to the nearest whole numberHigher: 14² = 196 and 15² = 225, so it lies between 14 and 15. 14.5² = 210.25 is above 200, so √200 ≈ 14.

Standard form: A × 10ⁿ with 1 ≤ A < 10 and n an integer. Large numbers have a positive index; numbers between 0 and 1 have a negative index.
Write 0.00943 and 45 000 000 in standard form: 0.00943 = 9.43 × 10⁻³ (point moves 3 places right). 45 000 000 = 4.5 × 10⁷.
Work out (3.2 × 10⁵) × (4 × 10⁻⁸): Multiply the numbers and add the indices: 12.8 × 10⁻³. Tidy A into 1 to 10: 1.28 × 10⁻².
Work out 4.5 × 10⁶ + 3.2 × 10⁵: Rewrite to the same power of 10: 4.5 × 10⁶ + 0.32 × 10⁶ = 4.82 × 10⁶. You do not add the indices when adding.
How do you order numbers in standard form?: Compare the powers of 10 first; when the powers match, compare the A values. Write every number in standard form first.

Perimeter and area of a semicircle of diameter 8, in terms of π: Perimeter = half of 8π + diameter = 4π + 8. Area = ½ × π × 4² = 8π.
Surd rulesHigher: √a × √b = √(ab) √a ÷ √b = √(a/b) √a × √a = a Roots do not split over addition: √(9 + 16) = 5, but √9 + √16 = 7.
Simplify √75Higher: Largest square factor is 25: √75 = √(25 × 3) = 5√3.
Rationalise 10/√5Higher: Multiply top and bottom by √5: 10√5/5 = 2√5.
Rationalise 3/(√7 + 2)Higher: Multiply top and bottom by the conjugate √7 − 2: 3(√7 − 2)/(7 − 4) = 3(√7 − 2)/3 = √7 − 2.