To solve the problem, we need to simplify the expression given and find the values of \( p \), \( q \), and \( r \) in the expression \( a^p b^q c^r \). Then, we will calculate \( p + 2q \). ### Step 1: Simplifying the Expression The expression we need to simplify is: \[ \frac{(a^2 b^{-3} c)^{3/4}}{a^{-1} b^4 c^5} \] #### Step 1.1: Simplifying the Numerator First, let's simplify the numerator \( (a^2 b^{-3} c)^{3/4} \): 1. **Apply the exponent**: When raising a product to a power, we distribute the exponent to each factor: \[ (a^2)^{3/4} (b^{-3})^{3/4} (c)^{3/4} = a^{2 \cdot \frac{3}{4}} b^{-3 \cdot \frac{3}{4}} c^{1 \cdot \frac{3}{4}} \] This simplifies to: \[ a^{\frac{3}{2}} b^{-\frac{9}{4}} c^{\frac{3}{4}} \] #### Step 1.2: Simplifying the Denominator Now, let's simplify the denominator \( a^{-1} b^4 c^5 \): - This is already in its simplest form: \[ a^{-1} b^4 c^5 \] ### Step 2: Combining the Numerator and Denominator Now we can combine the simplified numerator and denominator: \[ \frac{a^{\frac{3}{2}} b^{-\frac{9}{4}} c^{\frac{3}{4}}}{a^{-1} b^4 c^5} \] #### Step 2.1: Applying the Quotient Rule Using the quotient rule \( \frac{x^m}{x^n} = x^{m-n} \), we simplify each base: 1. For \( a \): \[ a^{\frac{3}{2} - (-1)} = a^{\frac{3}{2} + 1} = a^{\frac{3}{2} + \frac{2}{2}} = a^{\frac{5}{2}} \] 2. For \( b \): \[ b^{-\frac{9}{4} - 4} = b^{-\frac{9}{4} - \frac{16}{4}} = b^{-\frac{25}{4}} \] 3. For \( c \): \[ c^{\frac{3}{4} - 5} = c^{\frac{3}{4} - \frac{20}{4}} = c^{-\frac{17}{4}} \] ### Step 3: Final Expression Combining these results, we have: \[ \frac{(a^2 b^{-3} c)^{3/4}}{a^{-1} b^4 c^5} = a^{\frac{5}{2}} b^{-\frac{25}{4}} c^{-\frac{17}{4}} \] This can be expressed as: \[ a^{\frac{5}{2}} b^{-\frac{25}{4}} c^{-\frac{17}{4}} = a^p b^q c^r \] Where: - \( p = \frac{5}{2} \) - \( q = -\frac{25}{4} \) - \( r = -\frac{17}{4} \) ### Step 4: Calculating \( p + 2q \) 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 final answer 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 \( q \) and \( r \). - **Option B: \( -\frac{5}{4} \)**: This is also incorrect as it underestimates the negative impact of \( q \). - **Option C: \( -\frac{25}{4} \)**: This is incorrect because it does not correctly combine \( p \) and \( 2q \). ### Revision Summary - Simplify expressions using exponent rules and the quotient rule. - Combine like bases by subtracting exponents. - Carefully calculate \( p + 2q \) using the values derived from the simplified expression. - Always check each step to avoid common pitfalls in exponent manipulation.