Exponent Equation Simplification

The problem requires finding the value of '$m$' by simplifying the given exponential equation:

$(\frac{13}{10})^4 \times (\frac{10}{13})^{16} \times (\frac{13}{10})^{17} = (\frac{10}{13})^{3m + 18}$

Simplifying the Left Hand Side (LHS)

To simplify the LHS, we express all terms with a common base. We know that $(\frac{10}{13}) = (\frac{13}{10})^{-1}$. Substituting this into the equation:

$LHS = (\frac{13}{10})^4 \times ((\frac{13}{10})^{-1})^{16} \times (\frac{13}{10})^{17}$

Using the exponent rule $(a^x)^y = a^{x \times y}$:

$LHS = (\frac{13}{10})^4 \times (\frac{13}{10})^{-16} \times (\frac{13}{10})^{17}$

Using the exponent rule $a^x \times a^y = a^{x+y}$:

$LHS = (\frac{13}{10})^{4 + (-16) + 17}$

$LHS = (\frac{13}{10})^{5}$

Equating Bases and Solving for m

Now, the equation becomes:

$(\frac{13}{10})^{5} = (\frac{10}{13})^{3m + 18}$

To solve for '$m$', we need the bases to be the same. We can rewrite the LHS base:

$LHS = ((\frac{10}{13})^{-1})^{5} = (\frac{10}{13})^{-5}$

The equation is now:

$(\frac{10}{13})^{-5} = (\frac{10}{13})^{3m + 18}$

Since the bases are equal, the exponents must be equal:

$-5 = 3m + 18$

Now, solve for '$m$':

$3m = -5 - 18$

$3m = -23$

$m = -\frac{23}{3}$

Final Answer Calculation

The value of '$m$' that satisfies the equation is $-\frac{23}{3}$.