Problem Analysis

The question asks to find the value of 'm' by simplifying an exponential equation with fractional bases.

The given equation is:

$ \left(\frac{17}{4}\right)^9 \times \left(\frac{4}{17}\right)^8 \times \left(\frac{17}{4}\right)^{15} = \left(\frac{4}{17}\right)^{5m+15} $

Solving the Exponential Equation

To solve for 'm', we will use the laws of exponents, ensuring all terms have a common base.

• Step 1: Standardize the base. Convert the term $\left(\frac{4}{17}\right)^8$ to the base $\frac{17}{4}$. Since $\frac{4}{17} = \left(\frac{17}{4}\right)^{-1}$, we have: $ \left(\frac{4}{17}\right)^8 = \left( \left(\frac{17}{4}\right)^{-1} \right)^8 = \left(\frac{17}{4}\right)^{-8} $ The equation now becomes: $ \left(\frac{17}{4}\right)^9 \times \left(\frac{17}{4}\right)^{-8} \times \left(\frac{17}{4}\right)^{15} = \left(\frac{4}{17}\right)^{5m+15} $
• Step 2: Simplify the left-hand side (LHS). Apply the exponent rule $x^a \times x^b = x^{a+b}$ to combine the terms on the LHS: $ \text{LHS exponent} = 9 + (-8) + 15 = 1 + 15 = 16 $ So, the LHS is $\left(\frac{17}{4}\right)^{16}$.
• Step 3: Express the right-hand side (RHS) with the standard base. Convert the RHS $\left(\frac{4}{17}\right)^{5m+15}$ to the base $\frac{17}{4}$: $ \left(\frac{4}{17}\right)^{5m+15} = \left( \left(\frac{17}{4}\right)^{-1} \right)^{5m+15} = \left(\frac{17}{4}\right)^{-(5m+15)} $
• Step 4: Equate the exponents. The equation is now simplified to: $ \left(\frac{17}{4}\right)^{16} = \left(\frac{17}{4}\right)^{-(5m+15)} $ For the equality to hold, the exponents must be equal: $ 16 = -(5m+15) $
• Step 5: Solve the linear equation for m. $ 16 = -5m - 15 $ Add 15 to both sides: $ 16 + 15 = -5m $ $ 31 = -5m $ Divide by -5: $ m = \frac{31}{-5} $ $ m = -\frac{31}{5} $

Final Answer

The value of m satisfying the equation is $-\frac{31}{5}$.