Answer :
The wavelengths for frequencies of 5.00 x 1019 Hz and 4.00 x 109 Hz are 6 x 10-12 meters and 7.5 x 10-2 meters respectively, corresponding to gamma rays and microwaves in the electromagnetic spectrum. Therefore, the wavelengths are (a) 6 x 10-12 meters (which is in the gamma-ray range of the electromagnetic spectrum) and (b) 7.5 x 10-2 meters (which corresponds to the microwave part of the spectrum).
The wavelengths of electromagnetic waves in free space for given frequencies can be calculated using the formula wavelength (λ) = speed of light (c) / frequency (f). The speed of light in a vacuum is approximately 3.00 x 108 meters per second (m/s).
For frequency f = 5.00 x 1019 Hz, λ = 3.00 x 108 m/s / 5.00 x 1019 Hz = 6 x 10-12 meters.
For frequency f = 4.00 x 109 Hz, λ = 3.00 x 108 m/s / 4.00 x 109 Hz = 7.5 x 10-2 meters (or 75 centimeters).
Therefore, the wavelengths are (a) 6 x 10-12 meters (which is in the gamma-ray range of the electromagnetic spectrum) and (b) 7.5 x 10-2 meters (which corresponds to the microwave part of the spectrum).
