Standard Form
Write very large and very small numbers as a number between 1 and 10 times a power of ten, and calculate with them by hand and on a calculator.
What a learner can do afterwards
- Convert between ordinary numbers and standard form in both directions
- Multiply and divide in standard form using the index laws
- Correct an answer such as 34 × 10⁵ that is not yet in standard form
1 · Read
Standard form writes a number as a leader between 1 and 10 times a power of 10. 5,600,000 becomes 5.6 times 10^6 because the point moves 6 places. Small numbers use negative powers: 4.7 times 10^-4 is 0.00047.
Calculating splits into two easy jobs: leaders together, powers together. (2 times 10^3) times (3 times 10^4): 2 times 3 = 6, powers add 3 + 4 = 7, giving 6 times 10^7. Division mirrors it: (8 times 10^6) divided by (2 times 10^2) is 4 times 10^4.
If the leader drifts out of range, fix it before you finish. 34 times 10^5 is not standard form because 34 exceeds 10: split 34 into 3.4 times 10, then merge powers to get 3.4 times 10^6. The same repair rescues tiny leaders below 1. Always check the leader sits between 1 and 10.
Count places, not zeros, when you convert back: positive powers grow, negative powers shrink. This final tidy-up is where most marks hide, so never hand in a leader outside 1 to 10.
Leader between 1 and 10, powers add or subtract, then fix any drifting leader.
More to read
1.2 Exponents and Scientific Notation
Standard form writes big or tiny numbers as a number between 1 and 10 times a power of 10. So 5,600,000 becomes 5.6 times 10 to the 6, because the point moves 6 places. The leading number must stay in range.
Converting back means moving the point by the power: positive powers grow, negative powers shrink. For 4.7 times 10 to the minus 4, slide the point 4 places left to get 0.00047. Count places, not zeros.
Decimals: Rounding and Scientific Notation
Multiplying in standard form splits into two easy jobs: multiply the leading numbers, then add the powers. (2 times 10 cubed) times (3 times 10 to the 4) gives 6 times 10 to the 7. Index laws do the heavy lifting.
Division mirrors it: divide the leaders, subtract the powers. (8 times 10 to the 6) over (2 times 10 squared) is 4 times 10 to the 4. If the leader drifts out of range, adjust it and shift the power to compensate.
C | Scientific Notation
Answers like 34 times 10 to the 5 are NOT yet in standard form, because 34 exceeds 10. Fix by splitting 34 into 3.4 times 10, then merging powers: 3.4 times 10 to the 6. Always finish by checking the leader sits between 1 and 10.
The same repair works for tiny leaders: 0.09 times 10 cubed becomes 9 times 10 to the 1. Move one digit, shift the power by one. This final tidy up is where most marks hide.
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2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
17 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.