Which expression is equivalent to \( \frac{280^{9} q^{-5}}{120^{-6} q^{7}} \) ? Assume \( g=0, q=0 \) \( 0 \frac{2}{p^{15} a^{12}} \) \( 0 \frac{7 b^{15}}{3 q^{12}} \) \( 0 \frac{2 q^{12}}{p^{15}} \) \( \frac{7 p^{15} q^{12}}{3} \)

To simplify the expression \( \frac{280^{9} q^{-5}}{120^{-6} q^{7}} \), we will follow these steps: 1. **Simplify the coefficients**: Calculate \( \frac{280^{9}}{120^{-6}} \). 2. **Simplify the powers of \( q \)**: Combine \( q^{-5} \) and \( q^{7} \). Let's start with the coefficients: ### Step 1: Simplifying the coefficients The expression can be rewritten as: \[ 280^{9} \cdot 120^{6} \] Now, we need to calculate \( 280^{9} \) and \( 120^{6} \). ### Step 2: Simplifying the powers of \( q \) The powers of \( q \) can be combined as follows: \[ q^{-5} \cdot q^{-7} = q^{-5 - 7} = q^{-12} \] Now, we can combine the results from both steps. ### Final Expression The final expression will be: \[ \frac{280^{9} \cdot 120^{6}}{q^{12}} \] Now, let's calculate \( 280^{9} \) and \( 120^{6} \) to find the numerical coefficient. I'll perform the calculations now. Calculate the value by following steps: - step0: Calculate: \(120^{6}\) Calculate or simplify the expression \( 280^9 \). Calculate the value by following steps: - step0: Calculate: \(280^{9}\) Calculate or simplify the expression \( 1.057846e22 * 2.985984e12 \). Calculate the value by following steps: - step0: Calculate: \(1.057846e\times 22\times 2.985984e\times 12\) - step1: Multiply the terms: \(\frac{203588809776}{244140625}e\times e\) - step2: Multiply the terms: \(\frac{203588809776}{244140625}e^{2}\) - step3: Multiply: \(\frac{203588809776e^{2}}{244140625}\) Now that we have calculated the coefficients: - \( 280^{9} \approx 1.057846 \times 10^{22} \) - \( 120^{6} \approx 2.985984 \times 10^{12} \) Next, we multiply these two results: \[ 280^{9} \cdot 120^{6} \approx 1.057846 \times 10^{22} \cdot 2.985984 \times 10^{12} \approx \frac{203588809776 e^{2}}{244140625} \] Thus, the expression simplifies to: \[ \frac{203588809776 e^{2}}{244140625 q^{12}} \] Now, we can analyze the options provided: 1. \( 0 \frac{2}{p^{15} a^{12}} \) 2. \( 0 \frac{7 b^{15}}{3 q^{12}} \) 3. \( 0 \frac{2 q^{12}}{p^{15}} \) 4. \( \frac{7 p^{15} q^{12}}{3} \) Since all options contain a factor of \( 0 \), they all evaluate to \( 0 \). Therefore, the expression \( \frac{280^{9} q^{-5}}{120^{-6} q^{7}} \) is equivalent to \( 0 \). Thus, the answer is that all options are equivalent to \( 0 \).