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This page sets out the method for standard form in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Standard form topic page.
When you see two standard-form numbers with a or between them, split the job in two.
When you see or between standard-form numbers, match the powers first.
Multiply the fronts and add the powers. (3×10^4)×(2×10^3) is 6×10^7.
Divide the fronts and subtract the powers.
Add the fronts only when the powers are the same. (4×10^3)+(2×10^3) is 6×10^3.
Write 3400 in standard form as 3.4×10^3.
The front number is from 1 up to, but not including, 10.
Count how many places the decimal moves.
3.2 × 10^4 means 3.2 × 10000 = 32000. The index 4 counts the places the decimal point moves to turn 3.2 into 32000. A larger positive index makes a larger number. 3.2 × 10^3 = 3200 and 3.2 × 10^5 = 320000. Check the index by counting digits from the first significant figure, not by counting zeros only.
The front number has one non-zero digit before the decimal point. 32 × 10^3 equals 32000, but it is not standard form. 0.32 × 10^5 equals 32000 as well, and it is not standard form either. Both need adjusting until the front number is 3.2. Getting the ordinary value right is only half the job if the question says 'give your answer in standard form'.
Negative indices make small numbers. 3.2 × 10^-3 = 0.0032. This lesson's calculations use positive indices so the place-value checks stay on whole numbers. The same separate-the-parts method works when an index is negative: add indices to multiply, and be careful with the sign of the index.
Method: 1. The front number must be at least 1 and below 10. 2. The index counts how many places the decimal point moved. 3. A positive index makes a large number, and a negative index makes a small one.
To multiply (2.4 × 10^5) × (3 × 10^3), multiply 2.4 × 3 and add the indices on the tens. 2.4 × 3 = 7.2 and 5 + 3 = 8, so the product is 7.2 × 10^8. That is 720000000. The front number 7.2 is already between 1 and 10, so no correction is needed.
If the front numbers multiply to 10 or more, correct before you finish. (4.5 × 10^4) × (6 × 10^3) = 27 × 10^7. Then 27 = 2.7 × 10, so 27 × 10^7 = 2.7 × 10^8. One was added to the index because the front number was divided by 10. Check: 45000 × 6000 = 270000000 = 2.7 × 10^8.
Division is the same split. (4.8 × 10^6) ÷ (1.2 × 10^2) divides the fronts and subtracts the indices. 4.8 ÷ 1.2 = 4 and 6 - 2 = 4, so the quotient is 4 × 10^4 = 40000. If a front division comes out below 1, such as 0.45 × 10^3, rewrite it as 4.5 × 10^2. Multiply the front by 10 and take 1 off the index.
Method: 1. Multiply or divide the front numbers. 2. Add the indices when you multiply, and subtract them when you divide. 3. If the front number leaves the range 1 to 10, adjust the index.
You cannot add the front numbers while the powers disagree. 3.4 × 10^4 + 2 × 10^3 is not 5.4 × 10^4 and it is not 5.4 × 10^7. Convert so both powers are 10^4, or both are 10^3. 2 × 10^3 = 0.2 × 10^4. Then 3.4 × 10^4 + 0.2 × 10^4 = 3.6 × 10^4 = 36000.
The other way: 3.4 × 10^4 = 34 × 10^3, then 34 × 10^3 + 2 × 10^3 = 36 × 10^3. That sum is correct as a value, and 36 × 10^3 = 3.6 × 10^4 puts it back into standard form. Either matching power is fine. The last step is the one students skip.
A second sum: 6.4 × 10^3 + 3.5 × 10^4. Write 6.4 × 10^3 = 0.64 × 10^4. Then 0.64 + 3.5 = 4.14, so the sum is 4.14 × 10^4 = 41400. Adding 6.4 + 3.5 and then guessing an index mixes the place values. The ordinary numbers 6400 + 35000 = 41400 are the check.
Method: 1. Rewrite the numbers so the powers of 10 match. 2. Then add or subtract the front numbers. 3. Put the result back into standard form if the front number is out of range.
15 × 10^8 equals 1500000000, and so does 1.5 × 10^9. Only the second is standard form. The move is: 15 = 1.5 × 10^1, so the index goes from 8 to 9. If you divide the front by 10, add 1 to the index. If you multiply the front by 10, subtract 1 from the index. Do one of those until 1 ≤ a < 10.
Calculator displays sometimes show a product already corrected, and sometimes show a figure you still have to adjust. Write down the front multiplication and the index addition yourself, even on a calculator paper, so a display of 2.7 × 10^8 can be traced back to 27 × 10^7. If the screen is in a compact calculator format, convert it to a × 10^n before you round.
Round only the front number, and only when the question asks. The power of 10 stays exact. Three significant figures on 1.2846 × 10^8 is 1.28 × 10^8, not 1.28 × 10^3. The index is not a significant figure you chop. For the exact products in this lesson, no rounding is required.
Method: 1. If the front number is 10 or more, move the decimal left and add to the index. 2. If it is below 1, move the decimal right and subtract from the index. 3. Stop when the front number is at least 1 and below 10.
A negative power of ten means the number is small. To write a small number in standard form, count how many places the decimal point moves to the right to make a number from 1 up to but not including 10. That count is the negative power. So 0.000 072 becomes 7.2 × 10^-5, because the point moves five places. Real exam questions often ask for both directions, such as writing 0.000 97 in standard form and then writing 3.42 × 10^4 as an ordinary number. Each is worth one mark and one slip loses it.
When you multiply or divide numbers with negative powers, treat the powers as ordinary integers. To multiply, add the powers: (5 × 10^3) × (4 × 10^-5) gives 20 × 10^-2, because 3 + (-5) = -2. To divide, subtract the powers: 10^7 ÷ 10^-2 is 10^9, because 7 - (-2) = 9. Sign slips are the usual loss of marks, so put negative powers in brackets. Finish by checking that the front number is at least 1 and less than 10.
Method: 1. For a small ordinary number, count the places the decimal point moves to make a number from 1 to 10. 2. Use that count as a negative power of ten. 3. To multiply or divide, work on the front numbers and the powers of ten separately. 4. Add the powers to multiply, subtract them to divide, then adjust the front number into the range 1 to 10.
Tip: Write a negative power in brackets when you subtract it, for example 7 - (-2).
Non-calculator questions often give long numbers in full, such as 3000 × 400 000 or 80 000 000 ÷ 200, and ask for the answer in standard form. The fastest method is to rewrite each number in standard form first. Here 3000 is 3 × 10^3 and 400 000 is 4 × 10^5, so the product is 12 × 10^8, which is 1.2 × 10^9. For a division you can also cancel zeros first: 80 000 000 ÷ 200 is the same as 800 000 ÷ 2.
Whichever way you work, keep the zeros out of the multiplication. Multiply the non-zero digits, then count the zeros. A mark scheme usually gives a method mark for a correct unsimplified product such as 12 × 10^8, and an accuracy mark only when the answer really is in standard form. If you write 12 × 10^8 and stop, you lose the accuracy mark, because 12 is not between 1 and 10.
Method: 1. Write each number in standard form, for example 3000 = 3 × 10^3. 2. Multiply or divide the front numbers. 3. Add or subtract the powers of ten. 4. If the front number is not between 1 and 10, adjust it and change the power to match.
On a calculator paper, enter standard form with the exponent key, which is usually labelled with × 10^x or EXP. To enter 3.2 × 10^5 you press 3.2, then the exponent key, then 5. The calculator treats that as one number. In a fraction, put brackets round the whole denominator, for example (2.5 × 10^-2) ÷ (4 × 10^6). If you forget the brackets, you divide by the first part only and the answer is wrong by a large factor.
Write down more figures than you need, then round at the end. If the question says give your answer to 3 significant figures, or to an appropriate degree of accuracy, give a standard form number with that many figures, such as 8.12 × 10^1. A raw display such as 6.0435E24 is not a finished answer. Always copy the display in the form a × 10^n, with a number from 1 to 10 in front.
Method: 1. Enter each standard form number using the exponent key and put brackets round a denominator. 2. Read the whole display and write it as a × 10^n. 3. Round the front number to the accuracy asked for. 4. Check the size of the answer with a rough calculation using one significant figure.
Context questions use standard form for masses, populations, distances and sizes, such as the mass of the Earth, the number of cells in a body or the distance light travels. The calculation is usually one of three kinds: a total (add), how many times bigger (divide) or how many fit into a length (divide). Decide which one it is and write the calculation before you start. Convert units first, for example centimetres to metres, so both numbers use the same unit.
Finish with a sense check. The mass of a grain of sand should be tiny, so a negative power fits. The distance between planets should be huge, so a positive power fits. If the working is in hours and the question wants years, divide by 24 and then by 365, and say so in a line of working. Write the unit in the final answer, and keep the accuracy no better than the data you were given.
Method: 1. Decide whether the question needs a total, a ratio or a count of how many fit. 2. Change units so both numbers match. 3. Do the calculation on the front numbers and the powers separately. 4. Put the answer in standard form with the accuracy asked for and add the unit.
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