Standard Form flashcards
11 cards on Standard Form: the key points, worked examples and the slips to watch. Swipe through them, or print them.
Card 1 of 11
Standard Form
What standard form is
What is standard form, and what are the rules for the two parts?
Answer it in your head, then turn the card over.
What standard form is
Key points
- A number written as A × 10ⁿ, where A is at least 1 and less than 10.
- n is a whole number: positive for big numbers, negative for numbers between 0 and 1.
- So A always has exactly one non-zero digit before its decimal point.
Is it in standard form?
5.8 × 10⁷ is: 5.8 is between 1 and 10.
58 × 10⁶ has the same value, but 58 is too big, so it is not.
Both are 58 000 000.
Watch out
A must be less than 10, so 10 × 10⁴ is not standard form: it is 1 × 10⁵.
Standard Form
Large numbers
How do you write a large number in standard form, and change it back to an ordinary number?
Answer it in your head, then turn the card over.
Large numbers
Key points
- Move the decimal point left until the front number is between 1 and 10.
- The number of places it moved is the power of 10.
- To change back, move the point that many places right, filling gaps with zeros.
Example
63 400: the point moves 4 places to reach 6.34, so 6.34 × 10⁴
2.07 × 10⁵: move the point 5 places right to get 207 000
Watch out
Count the places the point moves, not the zeros: 2.07 × 10⁵ is 207 000, not 20 700 000.
Standard Form
Small numbers
How do you write a number less than 1 in standard form, and change it back?
Answer it in your head, then turn the card over.
Small numbers
Key points
- Move the decimal point right until the front number is between 1 and 10.
- The power is negative: minus the number of places the point moved.
- A negative power means dividing by 10 that many times.
- To change back, move the point that many places left.
Example
0.00038: the point moves 4 places to reach 3.8, so 3.8 × 10⁻⁴
6.1 × 10⁻³: move the point 3 places left to get 0.0061
Watch out
A number less than 1 never has a positive power: 0.00038 is 3.8 × 10⁻⁴, not 3.8 × 10⁴.
Standard Form
Ordering numbers in standard form
How do you put numbers written in standard form in order of size?
Answer it in your head, then turn the card over.
Ordering numbers in standard form
Key points
- Compare the powers of 10 first: the bigger power is the bigger number.
- Only when the powers are the same do you compare the front numbers.
- With negative powers, the most negative power is the smallest: 10⁻⁵ is less than 10⁻³.
Smallest first
8 × 10⁵ = 800 000
4.5 × 10⁶ = 4 500 000
1.2 × 10⁷ = 12 000 000
Watch out
A bigger front number does not beat a bigger power: 9 × 10⁶ is less than 2 × 10⁷.
Standard Form
Multiplying in standard form
How do you multiply two numbers written in standard form without a calculator?
Answer it in your head, then turn the card over.
Multiplying in standard form
Key points
- Multiply the front numbers.
- Add the powers of 10: 10³ × 10⁵ = 10⁸.
- If the front number comes out 10 or more, put it back into standard form.
(6 × 10⁴) × (7 × 10³)
6 × 7 = 42 and 4 + 3 = 7, so 42 × 10⁷
42 × 10⁷ = 4.2 × 10⁸
Watch out
Add the powers, never multiply them: 10⁴ × 10³ is 10⁷, not 10¹².
Standard Form
Dividing in standard form
How do you divide one number in standard form by another without a calculator?
Answer it in your head, then turn the card over.
Dividing in standard form
Key points
- Divide the front numbers.
- Subtract the powers of 10: the second power from the first.
- If the front number comes out less than 1, put it back into standard form.
(3 × 10⁵) ÷ (6 × 10⁻²)
3 ÷ 6 = 0.5 and 5 − (−2) = 7, so 0.5 × 10⁷
0.5 × 10⁷ = 5 × 10⁶
Watch out
Subtracting a negative power adds: 10⁵ ÷ 10⁻² is 10⁷, not 10³.
Standard Form
Fixing the front number
How do you rewrite an answer like 42 × 10⁷ or 0.5 × 10³ properly in standard form?
Answer it in your head, then turn the card over.
Fixing the front number
Key points
- The front number must be at least 1 and less than 10.
- Too big: divide it by 10 and add 1 to the power.
- Too small: multiply it by 10 and take 1 off the power.
- The value stays the same, because one part grows as the other shrinks.
Example
40 × 10⁶ = 4 × 10⁷
0.5 × 10³ = 5 × 10²
0.25 × 10⁻⁴ = 2.5 × 10⁻⁵
Watch out
Move the power the opposite way to the front number: 40 × 10⁶ is 4 × 10⁷, not 4 × 10⁵.
Standard Form
Standard form on a calculator
How do you enter standard form on a calculator, and write its answer down?
Answer it in your head, then turn the card over.
Standard form on a calculator
Key points
- Use the ×10ˣ key (EXP on some calculators): 2.4 ×10ˣ −3 enters 2.4 × 10⁻³.
- If you type × 10 ^ instead, put each number in brackets.
- Write the answer in proper standard form, as A × 10ⁿ.
(2.4 × 10⁻³) ÷ (8 × 10⁵)
The calculator gives 3 × 10⁻⁹.
Sense-check: 2.4 ÷ 8 = 0.3, and −3 − 5 = −8, so 0.3 × 10⁻⁸ = 3 × 10⁻⁹
Watch out
A display such as 3E−9 is not an answer to copy down: write it as 3 × 10⁻⁹.
Standard Form
Adding and subtracting in standard form
How do you add or subtract two numbers in standard form when their powers are different?
Answer it in your head, then turn the card over.
Adding and subtracting in standard form
Key points
- The front numbers only add when the powers of 10 match.
- Rewrite the smaller number with the bigger power: 8 × 10⁵ = 0.8 × 10⁶.
- Or change both to ordinary numbers, calculate, and change back.
- Put the answer back into standard form if it needs it.
5.3 × 10⁶ + 8 × 10⁵
= 5.3 × 10⁶ + 0.8 × 10⁶ = 6.1 × 10⁶
Check: 5 300 000 + 800 000 = 6 100 000
Watch out
Never add the fronts with different powers: 3.2 × 10⁵ + 4 × 10⁴ is 3.6 × 10⁵, not 7.2 × 10⁵.
Standard Form
How many times bigger
How do you answer a question asking how many times bigger one quantity is than another?
Answer it in your head, then turn the card over.
How many times bigger
Key points
- "How many times" is a division: the bigger value divided by the smaller.
- Divide the front numbers and subtract the powers, as usual.
- "How much bigger" is different: that one is a subtraction.
Bacterium 3 × 10⁻⁶ m, virus 1.2 × 10⁻⁷ m
(3 × 10⁻⁶) ÷ (1.2 × 10⁻⁷)
3 ÷ 1.2 = 2.5 and −6 − (−7) = 1, so 2.5 × 10¹
The bacterium is 25 times as long.
Watch out
Divide the bigger by the smaller: an answer below 1, like 0.04 here, means the division is upside down.
Standard Form
Units and sense-checking
How do you set up a standard form problem in context, such as how many cells fit in a length?
Answer it in your head, then turn the card over.
Units and sense-checking
Key points
- Put both quantities in the same units first: 1 cm = 10⁻² m.
- Set up the calculation, often a division, and keep the numbers in standard form.
- Sense-check: should the answer be huge or tiny?
Cells 8 × 10⁻⁶ m across, in 2 cm
2 cm = 2 × 10⁻² m
(2 × 10⁻²) ÷ (8 × 10⁻⁶) = 0.25 × 10⁴
= 2.5 × 10³, so 2500 cells
Watch out
Change cm to m first: dividing 2 by 8 × 10⁻⁶ gives 250 000, which is 100 times too many.
Print these cards
Four cards to an A4 sheet, each the size of a standard A6 revision card.
One-sided
Each card prints on one face: the question at the top, the notes under it.
Two-sided
Questions on the front, notes on the back. Print double-sided and flip on the long edge, then cut along the lines.
Put it into practice
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