Solving Exponential Equation for m

The problem asks to find the value of the variable $m$ that satisfies the given exponential equation:

$ \left(\frac{16}{5}\right)^{13} \times \left(\frac{5}{16}\right)^{16} \times \left(\frac{16}{5}\right)^{5} = \left(\frac{5}{16}\right)^{6m + 16} $

Step 1: Standardize the Base

To solve this equation efficiently, we need a common base for all terms. We observe that $\frac{5}{16}$ is the reciprocal of $\frac{16}{5}$. Using the property $\frac{a}{b} = \left(\frac{b}{a}\right)^{-1}$, we can rewrite $\frac{5}{16}$ as $\left(\frac{16}{5}\right)^{-1}$.

Substitute this into the original equation:

$ \left(\frac{16}{5}\right)^{13} \times \left(\left(\frac{16}{5}\right)^{-1}\right)^{16} \times \left(\frac{16}{5}\right)^{5} = \left(\left(\frac{16}{5}\right)^{-1}\right)^{6m + 16} $

Step 2: Apply Exponent Properties

Simplify the equation using the rules of exponents:

• Rule 1: $(a^x)^y = a^{xy}$. Apply this to the terms involving negative exponents.
• Rule 2: $a^x \times a^y = a^{x+y}$. Apply this to combine terms on the left side.

Applying Rule 1:

$ \left(\frac{16}{5}\right)^{13} \times \left(\frac{16}{5}\right)^{-1 \times 16} \times \left(\frac{16}{5}\right)^{5} = \left(\frac{16}{5}\right)^{-1 \times (6m + 16)} $

$ \left(\frac{16}{5}\right)^{13} \times \left(\frac{16}{5}\right)^{-16} \times \left(\frac{16}{5}\right)^{5} = \left(\frac{16}{5}\right)^{-(6m + 16)} $

Applying Rule 2 to the left side:

$ \left(\frac{16}{5}\right)^{13 + (-16) + 5} = \left(\frac{16}{5}\right)^{2} $

Step 3: Equate the Exponents

After simplification, the equation is:

$ \left(\frac{16}{5}\right)^{2} = \left(\frac{16}{5}\right)^{-(6m + 16)} $

Since the bases ($\frac{16}{5}$) are identical on both sides, the exponents must be equal:

$ 2 = -(6m + 16) $

Step 4: Solve the Linear Equation for m

Now, solve the resulting linear equation for $m$:

$ 2 = -6m - 16 $

Isolate the term with $m$ by adding 16 to both sides:

$ 2 + 16 = -6m $

$ 18 = -6m $

Finally, divide by -6 to find the value of $m$:

$ m = \frac{18}{-6} $

$ m = -3 $

Thus, the value of $m$ satisfying the equation is -3.