Ready to printMultiplying and dividing in scientific notation
The answer key prints on a separate page.
What's on this sheet
Multiplying two numbers in scientific notation is two moves that do not interfere with each other: multiply the coefficients, and add the exponents. Dividing is the same with the other two operations. A student who does both of those gets a true statement about two thirds of the time, and that is the problem — the other third is true and is not an answer.
Why this sheet works
5 × 10³ times 4 × 10² is 20 × 10⁵. That is exactly the right quantity, and it is not written in scientific notation, because the notation has one rule: the coefficient sits at 1 or above and below 10. The answer is 2 × 10⁶. Dividing has the mirror case, where the coefficient falls under 1 and the exponent goes down instead.
How this one works
One question from this sheet, worked through a step at a time.
(1.2 × 10⁻¹) ÷ (6 × 10³)
1.2 × 10⁻¹÷6 × 10³
coefficients1.2 ÷ 6powers10⁻¹ ÷ 10³
1.2÷6=0.2
−1−3=−4
0.2 × 10⁻⁴
is0.2outside 1 to 10
0.2 × 10⁻⁴
→2 × 10⁻⁵
(1.2 × 10⁻¹) ÷ (6 × 10³)=2 × 10⁻⁵
A number in scientific notation is two things multiplied: a coefficient and a power of ten. They travel separately — the coefficients divide with each other and the powers subtract with each other — so take them one at a time.
Divide the coefficients: 1.2 ÷ 6 = 0.2.
Now the powers. Dividing powers of the same base SUBTRACTS the exponents: −1 − 3 = −4. So the powers give 10⁻⁴.
Put them together: 0.2 × 10⁻⁴. Now CHECK it — a coefficient has to be at least 1 and less than 10, and 0.2 is not. This is a true number and it is not in scientific notation yet.
Move one factor of ten the other way: make the coefficient ten times BIGGER and the power ten times SMALLER. 0.2 × 10 = 2, and −4 − 1 = −5. The quantity has not changed — that is why the line above was true.
So the answer is 2 × 10⁻⁵ — 1.2 ÷ 6 = 0.2 and the powers give 10⁻⁴, and 0.2 is not between 1 and 10 — so it becomes 2 × 10⁻⁵.
