Solve.
$(6 \sqrt{6})^{-x+3}=\frac{1}{6} \cdot 216^{2 x-3}$
Enter your answer as a simplified fraction.
$x=$
We are given the exponential equation \((6 \sqrt{6})^{-x+3} = \frac{1}{6} \cdot 216^{2x-3}\). To solve for \(x\), we need to express both sides of the equation using the same base, which is 6. We know that \(6\sqrt{6} = 6^1 \cdot 6^{1/2} = 6^{3/2}\), \(216 = 6^3\), and \(\frac{1}{6} = 6^{-1}\). Substituting these into the original equation allows us to simplify it and compare the exponents.
First, rewrite the left-hand side:
\[(6 \sqrt{6})^{-x+3} = (6^{3/2})^{-x+3} = 6^{\frac{3}{2}(-x+3)} = 6^{-\frac{3}{2}x + \frac{9}{2}}\]
Next, rewrite the right-hand side:
\[\frac{1}{6} \cdot 216^{2x-3} = 6^{-1} \cdot (6^3)^{2x-3} = 6^{-1} \cdot 6^{3(2x-3)} = 6^{-1 + 6x - 9} = 6^{6x - 10}\]
Now, equate the two exponents:
\[-\frac{3}{2}x + \frac{9}{2} = 6x - 10\]
To solve the linear equation \(-\frac{3}{2}x + \frac{9}{2} = 6x - 10\), we first clear the fractions by multiplying every term by 2:
\[-3x + 9 = 12x - 20\]
Add \(3x\) to both sides and add \(20\) to both sides:
\[9 + 20 = 12x + 3x\]
\[29 = 15x\]
Dividing by 15 gives:
\[x = \frac{29}{15}\]
Exponential equations are used in real-world scenarios to model population growth, radioactive decay, or compound interest. For instance, if you have a bank investment that grows at a specific rate, you can set up an equation similar to this one to determine exactly when your investment will reach a target value. By expressing growth factors with a common base, you can efficiently calculate the time required for your money to grow as desired.