KBAT Problems
KBAT Problem: Working in Standard Form
A KBAT problem with very large or very small numbers where standard form keeps the arithmetic manageable and the answer sensible.
Keep the powers separate
When multiplying numbers in standard form, handle the number parts and the powers of ten separately, then combine. Keeping the answer in correct standard form, one non-zero digit before the point, is where the final mark lives.
Understand the problem
An original standard-form problem. Suppose one water molecule has a mass of 3.0 × 10⁻²³ g.
The question asks how many molecules are in 1.8 g of water. A very small number and an ordinary number appear together, so keeping them in standard form, one number part and one power of ten, stops the zeros from causing slips.
Plan and solve
- Number of molecules = total mass ÷ mass of one = 1.8 ÷ (3.0 × 10⁻²³).
- Divide the number parts: 1.8 ÷ 3.0 = 0.6.
- Subtract the indices: 10⁰ ÷ 10⁻²³ = 100−(−23) = 10²³.
- Combine: 0.6 × 10²³, then adjust to standard form = 6.0 × 10²² molecules.
Check and a variant
Check: 6.0 × 10²² × 3.0 × 10⁻²³ = 18 × 10⁻¹ = 1.8, back to the original mass, so the count is right. A twist: find the total mass of 5.0 × 10²⁵ molecules.
Multiply the number parts (5.0 × 3.0 = 15) and add the indices (10²⁵ × 10⁻²³ = 10²), giving 15 × 10² = 1.5 × 10³ = 1500 g, or 1.5 kg. The same two rules, number parts, then powers, handle both directions.
Frequently asked questions
When should I bother with standard form?
Whenever the numbers are very large or very small, so that counting zeros becomes error-prone. Writing 6.0 × 10²² instead of a twenty-three-digit number keeps the arithmetic short and the answer readable.
In the exam, if a value is given in standard form, keep working in it rather than expanding to ordinary notation.
What do I do with the powers of ten?
Handle them separately from the number parts. When you multiply, add the indices; when you divide, subtract them.
So 10⁰ ÷ 10⁻²³ gives 10²³. Keeping the powers in their own column stops the digits and the zeros from getting mixed up, which is where most mistakes happen.
Why change 0.6 × 10²³ to 6.0 × 10²²?
Because standard form needs the number part to be at least 1 but less than 10, and 0.6 is too small. Moving the decimal one place right makes it 6.0, balanced by lowering the power from 10²³ to 10²².
The value is unchanged; it is just written in the required form to earn the mark.