Question

1. \\(\left(\frac{10x^9}{x^{18}}\

ight)^{-5}\\)

Explanation:

Step1: Simplify the fraction inside the parentheses

First, simplify \(\frac{10x^{9}}{x^{18}}\) using the quotient rule of exponents \( \frac{a^m}{a^n}=a^{m - n} \). For the \(x\)-terms, we have \(x^{9-18}=x^{-9}\). So the fraction becomes \(10x^{-9}\).

Step2: Apply the power of a product rule and negative exponent rule

Now, apply the power \(-5\) to \(10x^{-9}\). Using the power of a product rule \((ab)^n=a^n b^n\), we get \(10^{-5}(x^{-9})^{-5}\). Then, using the power of a power rule \((a^m)^n=a^{mn}\), for the \(x\)-term: \((x^{-9})^{-5}=x^{(-9)\times(-5)} = x^{45}\). And \(10^{-5}=\frac{1}{10^{5}}=\frac{1}{100000}\).

Step3: Combine the results

Putting it together, we have \(\frac{1}{100000}x^{45}\) or \(\frac{x^{45}}{10^{5}}\) (which is \(\frac{x^{45}}{100000}\)).

Answer:

\(\frac{x^{45}}{100000}\) (or \(\frac{x^{45}}{10^{5}}\))