Question

1. \\(\frac{2x^{-2} cdot (2x)^0}{x^2}\\)

Explanation:

Step1: Simplify the zero exponent

Any non - zero number to the power of 0 is 1. So, \((2x)^{0}=1\) (assuming \(x
eq0\)). The expression becomes \(\frac{2x^{-2}\cdot1}{x^{2}}=\frac{2x^{-2}}{x^{2}}\)

Step2: Use the negative exponent rule

The negative exponent rule states that \(a^{-n}=\frac{1}{a^{n}}\), so \(x^{-2}=\frac{1}{x^{2}}\). Then the numerator \(2x^{-2}=\frac{2}{x^{2}}\). Now the expression is \(\frac{\frac{2}{x^{2}}}{x^{2}}\)

Step3: Divide by a fraction

Dividing by a fraction is the same as multiplying by its reciprocal. So \(\frac{\frac{2}{x^{2}}}{x^{2}}=\frac{2}{x^{2}}\cdot\frac{1}{x^{2}}\)

Step4: Multiply the fractions

When multiplying fractions, we multiply the numerators and the denominators. So \(\frac{2}{x^{2}}\cdot\frac{1}{x^{2}}=\frac{2}{x^{4}}\) (using the rule \(a^{m}\cdot a^{n}=a^{m + n}\), here \(x^{2}\cdot x^{2}=x^{2+2}=x^{4}\))

Answer:

\(\frac{2}{x^{4}}\)