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A physical quantity is expressed as f = 10^(x/(a*t) + 4), where x has dimensions of length and t has dimensions of time. For the exponent to be dimensionless, find the dimensional formula of the constant a.

1. [M⁰ L¹ T^(-1)]
2. [M⁰ L^(-1) T¹]
3. [M⁰ L^(1/2) T^(-1)]
4. [M L T^(-1)]

Correct answer: [M⁰ L¹ T^(-1)]

Solution

For 10^(x/(a*t)) to be a valid physical expression, the exponent x/(a*t) must be dimensionless. Dimensions: [x] = L, [t] = T. Dimensionless condition: [x]/([a]*[t]) = 1 -> [a] = [x]/[t] = L/T = [M⁰ L T^(-1)]. The constant 4 is already dimensionless. So [a] = [M⁰ L¹ T^(-1)].

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