Standard Form
Standard form (or scientific notation) writes very large or very small numbers as A × 10ⁿ, where A is between 1 and 10. The power n counts how many places the decimal point moves. So 0.007 becomes 7 × 10⁻³, a negative power.
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What Is Standard Form (Scientific Notation)?
- A way to write very large or very small numbers so they are easier to work with.
- It always looks like A × 10ⁿ, where A is between 1 and 10
How to Rewrite a Number into Standard Form?
- Move the decimal point so the number at the front is between 1 and 10.
- The number of places you move tells you the power of 10.
How the Power of 10 Works?
- Decimal moved left → positive power (e.g. 384000 → 3.84 × 10⁵)
- Decimal moved right → negative power (e.g. 0.007 → 7 × 10⁻³)
How to Multiply in Standard Form?
- Multiply the front numbers as normal.
- Add the powers of 10: 10⁷ × 10⁻¹⁻⁵
How Do We Check a Standard Form Answer?
- The front number must be between 1 and 10.
- If not, move the decimal and adjust the power (e.g. 12 × 10³ → 1.2 × 10⁴)
Practice Questions
Test your understanding
What is 25000 in standard form?
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Standard form is a number between 1 and 10 multiplied by a power of 10.
Move the decimal point in 25000 four places to the left to get 2.5, which is between 1 and 10. The point moved 4 places to the left, so multiply by 10⁴, giving 2.5 × 10⁴.
Count the places the decimal point moves, not the number of zeros, and always check the first number is between 1 and 10.
Simplify (2 × 10⁶) × (3 × 10⁻⁴). Give your answer in standard form.
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Multiply the front numbers: .
Add the indices of the powers of 10: 10⁶ × 10⁻⁴ ⁶⁺⁽⁻⁴⁾ .
So the answer is 6 × 10², which is already in standard form because 6 is between 1 and 10. .)
What is 0.0045 in standard form?
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Standard form is a number between 1 and 10 multiplied by a power of 10.
Move the decimal point in 0.0045 three places to the right to get 4.5, which is between 1 and 10. Moving the point to the right means the power is negative, so ⁻³.
The index is negative because the number is smaller than 1, not because the number itself is negative.
Write (5 × 10³) × (4 × 10⁻⁷) in standard form.
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Multiply the front numbers: .
Add the indices: 10³ × 10⁻⁷ ⁺⁽⁻⁷⁾ ⁻⁴. That gives 20 × 10⁻⁴.
20 is not between 1 and 10, so rewrite it as 2 × 10¹: 20 × 10⁻⁴ ¹ × 10⁻⁴ ⁻³ (which is 0.002). When you make the front number 10 times smaller, the power must go up by 1.
What is 670000 in standard form?
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Standard form is a number between 1 and 10 multiplied by a power of 10.
Move the decimal point in 670000 five places to the left to get 6.7, which is between 1 and 10. The point moved 5 places, so multiply by 10⁵, giving 6.7 × 10⁵.
Count the places the decimal point moves, not the zeros and not the digits.
Write (8.2 × 10⁶) × (2.5 × 10⁻⁴) in standard form.
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Multiply the front numbers: .
Add the indices: 10⁶ × 10⁻⁴ ⁶⁺⁽⁻⁴⁾ . That gives 20.5 × 10².
20.5 is not between 1 and 10, so rewrite it as 2.05 × 10¹: ¹ which is 2050). Making the front number 10 times smaller means the power goes up by 1.
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Standard form (scientific notation)
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Frequently asked questions
The questions students bump into most on this topic
Standard form writes a very large or very small number as A × 10ⁿ, where A is at least 1 and less than 10 and n is a whole number. So 384,000 is 3.84 × 10⁵ and 0.007 is 7 × 10⁻³. It keeps long strings of zeros out of calculations. In the US and on calculators it is called scientific notation.
Move the decimal point until you have a number between 1 and 10, then count the places you moved: that count is the power of 10. For 26,000, the point moves 4 places to the left to give 2.6, so 26,000 = 2.6 × 10⁴. In the same way 4,200 = 4.2 × 10³ and 543,000 = 5.43 × 10⁵.
Move the decimal point to the right until the number is between 1 and 10, and make the power negative. For 0.007 the point moves 3 places right to give 7, and each move made the number ten times bigger, so you balance it with 10⁻³: 0.007 = 7 × 10⁻³. In the same way, 0.000263 = 2.63 × 10⁻⁴. A negative power always means a number below 1.
Multiply the front numbers, then add the powers of 10. For 2 × 10⁷ times 8 × 10⁻¹²: 2 × 8 = 16 and 10⁷ × 10⁻¹² = 10⁻⁵, giving 16 × 10⁻⁵. But 16 is not between 1 and 10, so write it as 1.6 × 10¹ and add the powers again: 1.6 × 10⁻⁴. Forgetting that last step is the common mistake.
Divide the front numbers, then subtract the powers of 10. For 8 × 10⁷ divided by 2 × 10³: 8 ÷ 2 = 4 and 10⁷ ÷ 10³ = 10⁴, so the answer is 4 × 10⁴. If the front number ends up outside 1 to 10, adjust it as you would after multiplying: 2 × 10⁵ divided by 8 × 10² gives 0.25 × 10³, which is 2.5 × 10².
Write both numbers out in full, add or subtract as normal, then put the answer back into standard form. For 3 × 10⁴ + 5 × 10³: 30,000 + 5,000 = 35,000, which is 3.5 × 10⁴. You cannot add the powers, because adding is not multiplying; the powers only combine when they are the same.
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