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Cambridge IGCSE0580

Indices II

Mathematics 0580 Chapter Notes

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Indices II
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In this chapter

  1. 1Zero and negative indices
  2. 2Solving for the index
  3. 3Multiplying and dividing powers
  4. 4A power of a power
  5. 5Fractional indices
  6. 6Fractional powers of a product

1. Zero and negative indices

An index (or power) counts how many of a number are multiplied together: 3⁴ = 3 × 3 × 3 × 3.

Any non-zero number to the power 0 is 1. A negative index gives a reciprocal: the number goes under 1, and the index becomes positive. It does not make the answer negative: 2⁻³ is 1/8, not -8.

a⁰ = 1

a⁻ⁿ = 1/aⁿ

Examiner insight

“While this question was quite well answered, asking for a letter to the power of 0 did confuse quite a number of candidates.” “Part (b)(i) proved the most challenging of the simplifications with many not dealing with the negative power, hence writing the fraction upside down.”

Worked example 13 marks

Do not use a calculator. Work out (2/3)⁻² + 10⁰.

  1. 1

    negative index: invert the fraction, (3/2)² [1]

  2. 2

    square top and bottom: 9/4 [1]

  3. 3

    10⁰ = 1, so 9/4 + 1 = 13/4 [1]

2. Solving for the index

To solve an equation such as 2^(x) = 32, write both sides as a power of the same base. Here 32 = 2⁵, so x = 5.

When the bases match, the indices are equal. Then solve the equation that is left.

Do not work out a product such as 27 × 81: write each number as a power of 3, then add the indices.

a^(x) = a^(y) ⇒ x = y

Examiner insight

“This question proved quite challenging for a large proportion of candidates, with many multiplying 27 by 81 to get 2187 rather than looking to see if they could make the numbers have the same base as the unknown index number.” “While many candidates managed to write at least one of 9 and 27 as powers of 3, the step of multiplying them was usually a step too far.”

Worked example 13 marks

7^(2x+1) = 49^(2-x). Find the value of x.

  1. 1

    49 = 7², so 49^(2-x) = 7^(2(2-x)) = 7^(4-2x) [1]

  2. 2

    same base, so equate the indices: 2x + 1 = 4 - 2x [1]

  3. 3

    4x = 3, so x = 3/4 [1]

3. Multiplying and dividing powers

To multiply powers of the same letter, add the indices. To divide, subtract the indices.

Deal with the numbers separately from each letter. Multiply or divide the numbers, then work on each letter in turn.

A letter with no index has index 1. Subtracting a negative index adds: a⁵ ÷ a⁻² = a⁵⁻⁽⁻²⁾ = a⁷.

a^(m) × aⁿ = a^(m+n)

a^(m) ÷ aⁿ = a^(m-n)

Examiner insight

“The answer to this part was sometimes given as t3 as candidates divided the indices instead of subtracting.” “The most common incorrect answer was 7 from the misconception that the indices should be multiplied instead of added and most candidates who gave this answer also showed the calculation 28 ÷ 4.”

Worked example 12 marks

Do not use a calculator. Simplify 12a⁵ ÷ 3a⁻².

  1. 1

    divide the numbers: 12 ÷ 3 = 4 [1]

  2. 2

    subtract the indices: a⁵⁻⁽⁻²⁾ = a⁷, so 4a⁷ [1]

4. A power of a power

To raise a power to a power, multiply the indices: (x³)² = x⁶.

A power outside a bracket applies to every part inside it, the number as well as each letter. So (5x³)² = 5² × (x³)² = 25x⁶.

A negative power outside a bracket gives a reciprocal. Work out the number part completely, such as 10³ = 1000.

(a^(m))ⁿ = a^(mn)

(ab)ⁿ = aⁿ bⁿ

Examiner insight

“The biggest confusion was over applying the rules of indices to both a number and power.” “Part (b) was also well answered but answers of 9p6m15 and 3p6m15 were both seen quite often as candidates were unable to deal with the number correctly.”

Worked example 12 marks

Simplify. (10b⁴)⁻³

  1. 1

    apply the power to both parts: 10⁻³ × b^(4 × (-3)) = 10⁻³ b⁻¹² [1]

  2. 2

    negative index, reciprocal: 1/(1000 b¹²) [1]

5. Fractional indices

A fractional index is a root. The denominator is the root and the numerator is the power. So a^(1/2) is the square root, a^(1/3) is the cube root, and a^(2/3) is the cube root, then squared.

Take the root first, then the power. The numbers stay small and the working is exact. With a negative fractional index, take the reciprocal as well.

The answer to an index equation can be a fraction. For 32^(x) = 2, write 32 = 2⁵. Then 5x = 1, so x = 1/5.

a^(1/n) = √[n]a

a^(m/n) = (√[n]a)^(m)

Examiner insight

“Common errors included multiplying 27 by the fractional power.” “The less successful candidates did not consider the most sensible strategy for a non-calculator paper, and attempted to square 125 first, then take the cube root of the result.”

Worked example 14 marks

Do not use a calculator. Find the value of 27^(-2/3) × (6⁴)^(0.5).

  1. 1

    cube root first: 27^(1/3) = 3, so 27^(2/3) = 3² = 9 [1]

  2. 2

    negative index: 27^(-2/3) = 1/9 [1]

  3. 3

    (6⁴)^(0.5) = 6² = 36 [1]

  4. 4

    1/9 × 36 = 4 [1]

6. Fractional powers of a product

When a bracket holds a number and a power of a letter, apply the fractional index to both. Multiply the indices for the letter, and take the root of the number.

Then deal with a negative index by writing the answer as a fraction: the whole term goes under 1. Check the number and the letter both changed.

The rules for multiplying and dividing work the same way with fractional indices: x^(1/2) × x^(-3/4) = x^(-1/4).

Examiner insight

“Although many gave a correct answer, some applied the power to just the algebraic part of the term, giving the common partially correct answer of 64u30.”

Worked example 13 marks

Simplify. (27x⁹)^(-2/3)

  1. 1

    apply the power to both parts: 27^(-2/3) × (x⁹)^(-2/3) [1]

  2. 2

    27^(-2/3) = 1/9 and (x⁹)^(-2/3) = x⁻⁶ [1]

  3. 3

    1/9x⁶ [1]

Recap

  • a⁰ = 1 and a⁻ⁿ = 1/aⁿ. A negative index is a reciprocal, not a negative number.
  • Multiply powers: add the indices. Divide powers: subtract the indices.
  • Power of a power: multiply the indices. The power applies to the number and to every letter.
  • Solve a^(x) = a^(y) by writing both sides with the same base, then equating the indices.
  • Extended: a^(m/n) is the nth root of a, to the power m. Take the root first.
  • Extended: give a negative fractional power as a fraction, with the number and the letter both changed.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. Write 2⁻⁴ as a fraction.1 mark
  2. Do not use a calculator. Work out 7⁰ + 3⁻².2 marks
  3. Solve 5^(x+1) = 125.2 marks
  4. Simplify 15k³ ÷ 5k⁻⁴.2 marks
  5. Simplify 4p³q × 7p⁻⁵q².3 marks
  6. Simplify ((2x³)⁴)/8x⁵.3 marks
  7. Do not use a calculator. Work out 16^(-3/4).2 marks
  8. Do not use a calculator. Solve (1/4)^(x) = 32^(x-1).3 marks
  9. Do not use a calculator. A rectangle has area (16x¹⁰)^(3/4) cm² and length (4x⁻³)^(-1/2) cm. Find the width of the rectangle, in cm. Give your answer in the form kxⁿ.5 marks
  10. Do not use a calculator. A square has sides of length 8^(x) cm. A rectangle has length 32 cm and width 4^(x-1) cm. The square and the rectangle have the same area. Find the value of x.5 marks

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