If x is small,so that x^2 and higher powers c | vedclass

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(C) Given expression is $E = \frac{(1-2 x)^{-1}(1-3 x)^{-2}}{(1-4 x)^{-3}}$.Using the binomial expansion $(1+ax)^n \approx 1+nax$ for small $x$:$(1-2

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$1-4 x$

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(C) Given expression is $E = \frac{(1-2 x)^{-1}(1-3 x)^{-2}}{(1-4 x)^{-3}}$.Using the binomial expansion $(1+ax)^n \approx 1+nax$ for small $x$:$(1-2 x)^{-1} \approx 1 + (-1)(-2x) = 1 + 2x$.$(1-3 x)^{-2} \approx 1 + (-2)(-3x) = 1 + 6x$.$(1-4 x)^{-3} \approx 1 + (-3)(-4x) = 1 + 12x$.Substituting these into the expression:$E \approx \frac{(1+2x)(1+6x)}{(1+12x)} = \frac{1 + 6x + 2x + 12x^2}{1+12x}$.Neglecting $x^2$ and higher powers:$E \approx \frac{1+8x}{1+12x} = (1+8x)(1+12x)^{-1}$.Using the binomial expansion again:$E \approx (1+8x)(1-12x) = 1 - 12x + 8x - 96x^2$.Neglecting $x^2$:$E \approx 1 - 4x$.

If $x$ is small,so that $x^2$ and higher powers can be neglected,then the approximate value for $\frac{(1-2 x)^{-1}(1-3 x)^{-2}}{(1-4 x)^{-3}}$ is

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