Solving the Exponential Equation for x

We are asked to find the value of x in the equation: $81 \times \left(\frac{16}{25}\right)^{x+2} \div \left(\frac{3}{5}\right)^{2x+4} = 144$

Simplify the Equation

1. Rewrite terms with common bases. Notice that $81 = 3^4$ and $144 = 12^2 = (3 \times 4)^2 = 3^2 \times 4^2$. Also, $\frac{16}{25} = \left(\frac{4}{5}\right)^2$. Substitute these into the equation:

   $3^4 \times \left(\left(\frac{4}{5}\right)^2\right)^{x+2} \div \left(\frac{3}{5}\right)^{2x+4} = 3^2 \times 4^2$

2. Apply the power of a power rule ($(a^m)^n = a^{mn}$) and simplify the exponent:

   $3^4 \times \left(\frac{4}{5}\right)^{2(x+2)} \div \left(\frac{3}{5}\right)^{2x+4} = 3^2 \times 4^2$ $3^4 \times \left(\frac{4}{5}\right)^{2x+4} \div \left(\frac{3}{5}\right)^{2x+4} = 3^2 \times 4^2$

3. Combine the terms with the same exponent using the rule $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$. First, isolate the combined exponential term:

   $\left(\frac{4}{5}\right)^{2x+4} \div \left(\frac{3}{5}\right)^{2x+4} = \frac{3^2 \times 4^2}{3^4}$ $\left(\frac{4/5}{3/5}\right)^{2x+4} = \frac{4^2}{3^2}$ $\left(\frac{4}{3}\right)^{2x+4} = \left(\frac{4}{3}\right)^2$

Solve for x

1. Since the bases are the same ($\frac{4}{3}$), equate the exponents:

   $2x+4 = 2$

2. Solve the linear equation for x:

   $2x = 2 - 4$ $2x = -2$ $x = \frac{-2}{2}$ $x = -1$

The value of x is -1.