Quantity 𝑎 has dimensions of mass times length times current to the negative one. Quantity 𝑏 has dimensions of time times length to the negative one. What are the dimensions of 𝑎 divided by 𝑏?
The dimensions of this fraction 𝑎 divided by 𝑏 are equal to the dimensions of 𝑎 divided by the dimensions of 𝑏. In our problem statement, we’re given those dimensions of 𝑎 and 𝑏. As we’ve seen, 𝑀 represents mass, 𝐿 represents length, 𝐼 represents current, and 𝑇 represents time. Conventionally, we report the dimensions of a quantity all on one line rather than as a fraction as written here. Let’s algebraically rearrange this fraction then so that we can write it all on one line.
One way to do that is to take the dimensions in the denominator and move them to the numerator while changing the sign of their exponents. This means then that as we transfer the time 𝑇 to the numerator, it gains an exponent of negative one. And then for our length to the negative one dimension, that becomes length to the positive one, or simply length 𝐿. Written this way, notice that we have a length here as well as a length here. We can combine these to give the dimension of length squared.
We see that we’ve now written the dimensions of 𝑎 divided by 𝑏 all on one line. And we’ve done this by means of using negative exponents. The dimensions of 𝑎 divided by 𝑏 then are mass times length squared times time to the negative one times current to the negative one. These are the dimensions of this fraction.