To solve the equation \[ \frac{(a^2 b^{-3}c)^{\frac{3}{4}}}{a^{-1}b^{4}c^{5}}=a^{p} b^{q} c^{r} \] and find the value of \( p + 2q \), we will break down the problem step-by-step. ### Step 1: Simplify the Left Side First, we need to simplify the left side of the equation. We start with the numerator: \[ (a^2 b^{-3}c)^{\frac{3}{4}} \] Using the power of a product rule \((xy)^n = x^n y^n\), we can distribute the exponent \(\frac{3}{4}\): \[ = a^{2 \cdot \frac{3}{4}} b^{-3 \cdot \frac{3}{4}} c^{1 \cdot \frac{3}{4}} = a^{\frac{3}{2}} b^{-\frac{9}{4}} c^{\frac{3}{4}} \] Now, we can rewrite the entire left side: \[ \frac{(a^2 b^{-3}c)^{\frac{3}{4}}}{a^{-1}b^{4}c^{5}} = \frac{a^{\frac{3}{2}} b^{-\frac{9}{4}} c^{\frac{3}{4}}}{a^{-1} b^{4} c^{5}} \] ### Step 2: Simplify the Denominator Next, we simplify the denominator: \[ a^{-1} b^{4} c^{5} \] ### Step 3: Combine the Fractions Now we can combine the fractions: \[ = a^{\frac{3}{2} - (-1)} b^{-\frac{9}{4} - 4} c^{\frac{3}{4} - 5} \] Calculating each exponent: 1. For \(a\): \[ \frac{3}{2} - (-1) = \frac{3}{2} + 1 = \frac{3}{2} + \frac{2}{2} = \frac{5}{2} \] 2. For \(b\): \[ -\frac{9}{4} - 4 = -\frac{9}{4} - \frac{16}{4} = -\frac{25}{4} \] 3. For \(c\): \[ \frac{3}{4} - 5 = \frac{3}{4} - \frac{20}{4} = -\frac{17}{4} \] Putting it all together, we have: \[ \frac{(a^2 b^{-3}c)^{\frac{3}{4}}}{a^{-1}b^{4}c^{5}} = a^{\frac{5}{2}} b^{-\frac{25}{4}} c^{-\frac{17}{4}} \] ### Step 4: Identify \(p\), \(q\), and \(r\) From the equation \[ a^{\frac{5}{2}} b^{-\frac{25}{4}} c^{-\frac{17}{4}} = a^{p} b^{q} c^{r} \] we can identify: - \(p = \frac{5}{2}\) - \(q = -\frac{25}{4}\) - \(r = -\frac{17}{4}\) ### Step 5: Calculate \(p + 2q\) Now we need to calculate \(p + 2q\): \[ p + 2q = \frac{5}{2} + 2\left(-\frac{25}{4}\right) \] Calculating \(2q\): \[ 2q = 2 \cdot -\frac{25}{4} = -\frac{50}{4} = -\frac{25}{2} \] Now, substituting back into the equation: \[ p + 2q = \frac{5}{2} - \frac{25}{2} = \frac{5 - 25}{2} = \frac{-20}{2} = -10 \] ### Conclusion Thus, the value of \(p + 2q\) is \[ \boxed{-10} \] ### Explanation of Other Options - **Option A: \(\frac{5}{2}\)** - This is incorrect because it does not account for the negative contributions from \(q\). - **Option B: \(-\frac{5}{4}\)** - This is incorrect as it underestimates the negative impact of \(q\). - **Option C: \(-\frac{25}{4}\)** - This is incorrect as it miscalculates the contributions from both \(p\) and \(q\). ### Revision Summary - Simplify expressions using exponent rules. - Combine fractions by subtracting exponents. - Identify coefficients \(p\), \(q\), and \(r\) from the simplified expression. - Carefully calculate \(p + 2q\) to avoid arithmetic errors.