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To show that the equation = (h)/(mv) is homogeneous, we need to verify that the units on both sides of the equation are the same. Step 1: Identify the units of the physical quantities. • is wavelength, its unit is meter (m). • h is Planck's constant, its unit is Joule-second (J·s). • m is mass, its unit is kilogram (kg). • v is velocity, its unit is meter per second (m/s). Step 2: Express the unit of Planck's constant (h) in fundamental SI units (kg, m, s). The unit of energy, Joule (J), is defined as the work done by a force of 1 Newton over a distance of 1 meter. 1 J = 1 N · m Force (Newton, N) is defined by Newton's second law (F=ma): 1 N = 1 kg · m/s^2 So, the unit of Joule is: 1 J = (1 kg · m/s^2) · m = 1 kg · m^2/s^2 Therefore, the unit of Planck's constant h is: [h] = J · s = (kg · m^2/s^2) · s = kg · m^2/s Step 3: Calculate the units of the right-hand side of the equation ((h)/(mv)). The units of the numerator are: [h] = kg · m^2/s The units of the denominator are: [m] · [v] = kg · (m/s) Now, we divide the units of the numerator by those of the denominator: [(h)/(mv)] = kg · m^2/skg · m/s Step 4: Simplify the units of the right-hand side. [(h)/(mv)] = kg · m^2s × skg · m [(h)/(mv)] = kg · m^2 · ss · kg · m = m Step 5: Compare the units of both sides. The units of the left-hand side of the equation ([]) are meters (m). The units of the right-hand side of the equation ([(h)/(mv)]) are also meters (m). Since the units on both sides of the equation are identical (meters), the equation is homogeneous. The equation is homogeneous. That's 2 down. 3 left today — send the next one.