Evaluate (0.02)² × (0.03)³.

Step-by-step solution

1. 1

   Separate the digits from the decimal places. The reliable rule for multiplying decimals is: multiply as if the numbers were whole, then give the answer as many decimal places as the two factors have between them. Powers make the bookkeeping easy if you track each separately.

2. 2

   Handle the first power. $0.02$ has $2$ decimal places, so squaring gives $2\times 2=4$ decimal places, and $2^{2}=4$:

   $(0.02)^{2}=0.0004$

3. 3

   Handle the second power. $0.03$ has $2$ decimal places, so cubing gives $3\times 2=6$ decimal places, and $3^{3}=27$:

   $(0.03)^{3}=0.000027$

4. 4

   Multiply the two results. Digits: $4\times 27=108$. Decimal places: $4+6=10$. So the answer has $108$ preceded by enough zeros to sit $10$ places after the point:

   $0.0004\times 0.000027=0.0000000108$

5. 5

   Express in scientific notation.

   $0.0000000108=1.08\times 10^{-8}$

   $\boxed{1.08\times 10^{-8}}$

6. 6

   Check with powers of ten. Write $0.02=2\times 10^{-2}$ and $0.03=3\times 10^{-2}$. Then

   $\left(2\times 10^{-2}\right)^{2}\left(3\times 10^{-2}\right)^{3}=\left(4\times 10^{-4}\right)\left(27\times 10^{-6}\right)=108\times 10^{-10}=1.08\times 10^{-8}$

   which matches ✓. Scientific notation is the safest route once more than a handful of zeros are involved.