To convert the number [tex]\( 5983.6 \times 10^{-5} \)[/tex] into scientific notation, follow these steps:
1. Identify the given number and its exponent:
The number provided is [tex]\( 5983.6 \)[/tex] and the exponent is [tex]\( -5 \)[/tex].
2. Express the number in a typical scientific notation form:
Scientific notation requires the coefficient (significant figure) to be a number between 1 and 10. Thus, we need to transform [tex]\( 5983.6 \)[/tex] into such a coefficient.
3. Re-write the original number to fit within the common form [tex]\( d.ddd \times 10^y \)[/tex]:
Move the decimal point in [tex]\( 5983.6 \)[/tex] three places to the left to get [tex]\( 5.9836 \)[/tex]:
[tex]\[
5983.6 = 5.9836 \times 10^3
\][/tex]
4. Combine the exponents:
The original exponent is [tex]\( -5 \)[/tex]. Since we expressed [tex]\( 5983.6 \)[/tex] as [tex]\( 5.9836 \times 10^3 \)[/tex], we need to adjust the exponent [tex]\( -5 \)[/tex] by adding the [tex]\( 3 \)[/tex] from the shift in the decimal point. Thus:
[tex]\[
10^{-5} \text{ combined with } 10^3 \text{ gives us } 10^{-5 + 3} = 10^{-2}
\][/tex]
5. Final scientific notation:
Therefore, the expression [tex]\( 5983.6 \times 10^{-5} \)[/tex] in scientific notation is:
[tex]\(
5.9836 \times 10^{-2}
\)[/tex]
So, placing the coefficient in the green box and the exponent in the yellow box:
[tex]\[ \boxed{\textcolor{green}{5.9836}} \times 10^{\boxed{\textcolor{yellow}{-2}}} \][/tex]
