If `Q=(x^n)/(y^m) and Deltax, Deltay` are absolute errors in the measurement of x and y then absolute error `DeltaQ ` in Q is

Options

A. `DeltaQ =pm (n(Deltax)/x+m(Deltay)/y)`
B. `DeltaQ=pm(n(Deltax)/x+m(Deltay)/y)Q`
C. `DeltaQ=pm(n(Deltax)/x-m(Deltay)/y)Q`
D. `DeltaQ=pm(n(Deltax)/y-m(Deltay)/x)Q`

Answer

To find the absolute error \(\Delta Q\) in the quantity \(Q = \frac{x^n}{y^m}\), we can follow these steps:

### Step 1: Define the quantity \(Q\)
Given:
\[ Q = \frac{x^n}{y^m} \]

### Step 2: Use the formula for relative error
The relative error in \(Q\) can be expressed in terms of the relative errors in \(x\) and \(y\). The formula for the relative error in a quotient is:
\[ \frac{\Delta Q}{Q} = n \frac{\Delta x}{x} + m \frac{\Delta y}{y} \]
where \(\Delta x\) and \(\Delta y\) are the absolute errors in \(x\) and \(y\) respectively.

### Step 3: Rearranging the equation
From the relative error equation, we can express the absolute error \(\Delta Q\) as:
\[ \Delta Q = Q \left( n \frac{\Delta x}{x} + m \frac{\Delta y}{y} \right) \]

### Step 4: Substitute \(Q\) back into the equation
Now, substituting \(Q\) back into the equation:
\[ \Delta Q = \frac{x^n}{y^m} \left( n \frac{\Delta x}{x} + m \frac{\Delta y}{y} \right) \]

### Step 5: Simplify the expression
This can be simplified to:
\[ \Delta Q = \frac{x^n}{y^m} \left( n \frac{\Delta x}{x} + m \frac{\Delta y}{y} \right) = n \frac{x^{n-1} \Delta x}{y^m} + m \frac{x^n \Delta y}{y^{m+1}} \]

### Final Result
Thus, the absolute error \(\Delta Q\) in \(Q\) is given by:
\[ \Delta Q = Q \left( n \frac{\Delta x}{x} + m \frac{\Delta y}{y} \right) \]