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Cambridge IGCSE0580

Standard form

Mathematics 0580 Chapter Notes

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Standard form
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In this chapter

  1. 1What standard form is
  2. 2Converting into and out of standard form
  3. 3Multiplying and dividing
  4. 4Adding and subtracting
  5. 5Powers and roots

1. What standard form is

A number is in standard form when it is written as A × 10ⁿ, where n is a positive or negative integer and 1 leqslant A < 10.

So A has exactly one non-zero digit in front of the decimal point. A number of 10 or more has a positive power of 10. A number less than 1 has a negative power of 10.

A × 10ⁿ, 1 leqslant A < 10, n an integer

Examiner insight

“While this standard form question was well answered, quite a few candidates did not know that the first part of this form had to be a value between 1 and 10.”

Worked example 12 marks

710 000.03 = (7.1 × 10^(a)) + (3 × 10^(b))

Find the value of a and the value of b.

  1. 1

    710 000 = 7.1 × 10⁵, so a = 5 [1]

  2. 2

    0.03 = 3 × 10⁻², so b = -2 [1]

2. Converting into and out of standard form

To convert into standard form, move the decimal point until there is one non-zero digit in front of it. Count the places moved. Moving it left, for a number of 10 or more, gives a positive power. Moving it right, for a number less than 1, gives a negative power.

To convert out of standard form, move the decimal point the other way by n places.

6320 = 6.32 × 10³

0.0258 = 2.58 × 10⁻²

7.83 × 10⁻² = 0.0783

Examiner insight

“The most common error was for candidates to replace the power of –3 with +3” “The most frequent error was to give a power of –5 rather than 5.”

Worked example 12 marks

Write these numbers in standard form.

(a) 6320

(b) 0.0258

  1. 1

    (a) 6320 = 6.32 × 10³ [1]

  2. 2

    (b) 0.0258 = 2.58 × 10⁻² [1]

3. Multiplying and dividing

Multiply or divide the A parts. Then use the laws of indices on the powers of 10: add the powers to multiply, subtract them to divide.

If the A part of the answer is 10 or more, or less than 1, adjust it. The answer must finish in standard form.

A calculator may display the answer as an ordinary number. Write it in standard form yourself.

10^(a) × 10^(b) = 10^(a+b)

10^(a) ÷ 10^(b) = 10^(a-b)

Examiner insight

“This needs to be manually turned into a standard form number. Often this second stage was not attempted.”

Worked example 12 marks

Work out.

(5.5 × 10⁶) × (2 × 10⁻³)

Give your answer in standard form.

  1. 1

    5.5 × 2 = 11 and 10⁶ × 10⁻³ = 10³, so 11 × 10³ [1]

  2. 2

    11 × 10³ = 1.1 × 10⁴ [1]

4. Adding and subtracting

You cannot add or subtract the A parts until both numbers have the same power of 10. Rewrite one number so its power matches the other. Then add or subtract the A parts and keep the power.

Check that the answer is in standard form.

4.2 × 10³¹ = 0.42 × 10³²

Examiner insight

“Many correctly amended the terms so that the powers of 10 were the same value in order to answer this question.” “Quite a few candidates made arithmetic errors when subtracting 0.42 from 4.2.”

Worked example 12 marks

Work out (4.2 × 10³²) - (4.2 × 10³¹).

Give your answer in standard form.

  1. 1

    4.2 × 10³¹ = 0.42 × 10³² [1]

  2. 2

    4.2 - 0.42 = 3.78, so the answer is 3.78 × 10³² [1]

5. Powers and roots

To raise a standard form number to a power, raise the A part to that power and multiply the power of 10 by it. A root works the same way: take the root of both parts.

Before taking a root, rewrite the number so that both parts can be rooted exactly.

(A × 10ⁿ)^(k) = A^(k) × 10^(nk)

Examiner insight

“Most candidates worked with the standard form number, squaring the 4 and doubling the power of 10 before adjusting the answer to standard form.” “Most candidates did not realise they needed to amend the terms to allow cube rooting to take place.”

Worked example 12 marks

Work out (80 × 10⁵⁰)^(1/3).

Give your answer in standard form.

  1. 1

    80 × 10⁵⁰ = 8 × 10⁵¹ [1]

  2. 2

    cube root of each part: √[3]8 = 2 and 10^(51 ÷ 3) = 10¹⁷, so 2 × 10¹⁷ [1]

Recap

  • Standard form is A × 10ⁿ with 1 leqslant A < 10 and n an integer.
  • Numbers of 10 or more have a positive power. Numbers less than 1 have a negative power.
  • Multiply: multiply the A parts and add the powers. Divide: divide the A parts and subtract the powers.
  • Add or subtract: make the powers of 10 the same first.
  • Always check the final answer has A at least 1 and less than 10.
  • Extended: powers and roots act on both parts. Rewrite first so both parts can be rooted exactly.

End-of-chapter exercise

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

  1. Do not use a calculator. Which of these is written in standard form? 24 × 10³, 0.7 × 10⁵, 3.6 × 10⁻⁵, 10 × 10²1 mark
  2. Do not use a calculator. Write 0.00704 in standard form.1 mark
  3. Do not use a calculator. Write 3.06 × 10⁻⁴ as an ordinary number.1 mark
  4. The mass of a speck of dust is 63 × 10⁻⁶ g. Write this mass in standard form.2 marks
  5. Calculate (3 × 10⁵) × (4 × 10³). Give your answer in standard form.2 marks
  6. Calculate (2.8 × 10⁶) + (5.5 × 10⁵). Give your answer in standard form.2 marks
  7. Do not use a calculator. Work out (6.4 × 10⁷) ÷ (1.6 × 10⁻²). Give your answer in standard form.2 marks
  8. Do not use a calculator. Work out (5 × 10⁻³)². Give your answer in standard form.2 marks
  9. Do not use a calculator. Work out √((4.3 × 10⁻⁷) - (7 × 10⁻⁸)). Give your answer in standard form.4 marks
  10. Do not use a calculator. A colony contains 8 × 10⁹ bacteria. Each bacterium has a mass of 9.5 × 10⁻¹³ g. A second colony has a total mass of 6 × 10⁻⁴ g. Work out the total mass, in grams, of the two colonies. Give your answer in standard form.4 marks

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