Scientific notation question involving multiplication of factors with different power

Question

(5.32 x10(8)) * (3.54 x 10(3)) I do this by converting 3.54 to a power of 8 so:

(5.32 x10(8)) * (.0000354 x10(8)) which, when I do the math, equals .0000188328 or 1.88328 x10(-4)

HOWEVER

The answer apparently is 1.88328 x10(12)

Where / what have I messed up?

Answer 1

You should have added the exponents as well:

$$ (a \times 10^8) \, (b \times 10^8) = (ab)\times 10^{8+8} = (ab)\times 10^{16} $$

Answer 2

Making the powers of 10 agree is a detour. You need that for addition, but not for multiplication. In multiplication you simply add the exponents:

$$ \begin{align} (5.32\times 10^8) \times (3.54 \times 10^3) =& (5.32 \times 3.54) \times (10^8 \times 10^3) \\=& 18.8328 \times 10^{8+3} \\=& (1.88328 \times 10^1) \times 10^{11} \\=& 1.88328 \times 10^{12} \end{align}$$