To second what @scottdave said, the exponent must be dimensionless. Also the argument to a sin or cos or other trig function or a log must all be dimensionless.Nikhil faraday said:Must two quantities have the same dimensions if you are using one quantity as an exponent in raising other to a power?
Dimensions in science refer to the physical units or measurements used to describe a quantity, such as length, time, mass, or temperature. These dimensions are typically represented using standard units, such as meters, seconds, kilograms, or degrees Celsius.
In order for two quantities to be compared or combined in a meaningful way, they must have the same dimensions. This is because quantities with different dimensions cannot be added, subtracted, multiplied, or divided. In other words, the dimensions must be consistent for calculations to be valid.
If two quantities with different dimensions are added together, the result will be a nonsensical value. For example, adding 5 meters to 10 seconds would result in 15 meters-seconds, which does not have a real-world meaning. This is why it is important for quantities to have the same dimensions when performing calculations.
No, dimensions can vary depending on the system of units used. For example, length can be measured in meters, feet, or inches, and each will have a different numerical value. However, the dimensions will always be the same - in this case, length - regardless of the unit used.
Yes, dimensions can be converted between different units as long as they are the same type of measurement. For example, length can be converted from meters to centimeters by multiplying by 100. However, dimensions cannot be converted between different types of measurements, such as length and time.