Question

1. simplify ((2d)^0 div (d^2 e^{-1})^{-1}).

Explanation:

Step1: Apply the zero - exponent rule

Recall that any non - zero number raised to the power of 0 is 1. So, for \((2d)^0\), since \(2d\) is non - zero (assuming \(d
eq0\)), we have \((2d)^0 = 1\).

Step2: Simplify the denominator using the power of a product and negative exponent rules

First, simplify \((d^{2}e^{-1})^{-1}\). Using the power of a product rule \((ab)^n=a^{n}b^{n}\) and the negative exponent rule \((a^{m})^{n}=a^{mn}\), we get:
\((d^{2}e^{-1})^{-1}=(d^{2})^{-1}(e^{-1})^{-1}\)
Using the power - of - a - power rule \((a^{m})^{n}=a^{mn}\), we have \((d^{2})^{-1}=d^{-2}\) and \((e^{-1})^{-1}=e^{1}=e\). So, \((d^{2}e^{-1})^{-1}=d^{-2}e\)

Step3: Perform the division

We now have to divide 1 by \(d^{-2}e\). Recall that dividing by a fraction (or an expression) is the same as multiplying by its reciprocal. So, \(1\div(d^{-2}e)=\frac{1}{d^{-2}e}\)
Using the negative exponent rule \(a^{-n}=\frac{1}{a^{n}}\) (or \(\frac{1}{a^{-n}} = a^{n}\)), we can rewrite \(\frac{1}{d^{-2}e}\) as \(\frac{d^{2}}{e}\)

Answer:

\(\frac{d^{2}}{e}\)