### Related Formula By the Principle of Homogeneity, terms added or subtracted must have the same dimensions: [B] = [x^2]$[B] = [x^2]$ ### Core Logic Known dimensional formulas: Length x to [L]$x \to [L]$ Energy E to [ML^2T^-2]$E \to [ML^2T^{-2}]$ Time t to [T]$t \to [T]$ ### Step 1: Dimension of B Since x^2$x^2$ is subtracted from B$B$: [B] = [x^2] = [L^2]$[B] = [x^2] = [L^2]$ ### Step 2: Dimension of A From the equation E = fracB - x^2At$E = \frac{B - x^2}{At}$: [A] = frac[B - x^2][E][t]$[A] = \frac{[B - x^2]}{[E][t]}$ [A] = frac[L^2][ML^2T^-2][T] = frac[L^2][ML^2T^-1]$[A] = \frac{[L^2]}{[ML^2T^{-2}][T]} = \frac{[L^2]}{[ML^2T^{-1}]}$ [A] = [M^-1T^1]$[A] = [M^{-1}T^1]$ ### Step 3: Dimension of AB [AB] = [A] times [B]$[AB] = [A] \times [B]$ [AB] = [M^-1T^1] times [L^2]$[AB] = [M^{-1}T^1] \times [L^2]$ [AB] = [L^2 M^-1 T^1]$[AB] = [L^2 M^{-1} T^1]$ ### Pattern Recognition Identify sums/differences first to instantly isolate B$B$. Once [B]$[B]$ is fixed, the entire numerator is just L^2$L^2$. Swap out variables to isolate [A]$[A]$. Combining is just standard exponent addition. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Physics: Units and Measurements