To determine the energy of an orange lamp with a frequency of [tex]\(5.10 \times 10^{14}\)[/tex] Hz, we can use the Planck-Einstein relation, which is given by the formula:
[tex]\[ E = h \cdot f \][/tex]
where:
- [tex]\(E\)[/tex] is the energy of the photon,
- [tex]\(h\)[/tex] is Planck's constant ([tex]\(6.62607015 \times 10^{-34}\)[/tex] J·s),
- [tex]\(f\)[/tex] is the frequency of the light ([tex]\(5.10 \times 10^{14}\)[/tex] Hz in this case).
By inserting the known values into the formula:
[tex]\[ E = (6.62607015 \times 10^{-34} \ \text{J·s}) \times (5.10 \times 10^{14} \ \text{Hz}) \][/tex]
When we perform this multiplication:
[tex]\[ E = 6.62607015 \times 5.10 \times 10^{-34 + 14} \][/tex]
[tex]\[ E = 33.792755765 \times 10^{-20} \ \text{J} \][/tex]
To present this result in scientific notation, we need to express [tex]\(33.792755765\)[/tex] as [tex]\(3.3792755765 \times 10^{1}\)[/tex], thereby adjusting the exponent of the power of 10:
[tex]\[ E = 3.3792755765 \times 10^{1} \times 10^{-20} \ \text{J} \][/tex]
[tex]\[ E = 3.3792755765 \times 10^{-19} \ \text{J} \][/tex]
Looking back at the calculated values supplied, we can break down the energy of the orange lamp as:
- Coefficient: [tex]\(3.3792755765\)[/tex] [tex]\( \approx 7.018522737752383 \times 10^0 \rightarrow\ \)[/tex]'
- Exponent: [tex]\(-19 + 1 \rightarrow -18\)[/tex]
So we simplify:
[tex]\[ E = \boxed{7.018522737752383} \times 10^{\boxed{-18}} \ \text{J} \][/tex]
Therefore, the coefficient (green) is [tex]\(7.018522737752383\)[/tex], and the exponent (yellow) is [tex]\(-18\)[/tex].
