Let's solve the equation \( \left(\frac{11}{9}\right)^3 \times \left(\frac{9}{11}\right)^6 = \left(\frac{11}{9}\right)^{2n-1} \).
First, observe the left-hand side of the equation:
[tex]\[ \left(\frac{11}{9}\right)^3 \times \left(\frac{9}{11}\right)^6 \][/tex]
Rewriting \( \left(\frac{9}{11}\right)^6 \) in terms of \( \left(\frac{11}{9}\right) \), we get:
[tex]\[ \left(\frac{9}{11}\right)^6 = \left(\frac{11}{9}\right)^{-6} \][/tex]
Substitute this back into the equation:
[tex]\[ \left(\frac{11}{9}\right)^3 \times \left(\frac{11}{9}\right)^{-6} \][/tex]
Using the property of exponents \( a^m \times a^n = a^{m+n} \):
[tex]\[ \left(\frac{11}{9}\right)^{3 + (-6)} = \left(\frac{11}{9}\right)^{-3} \][/tex]
Now, we have the simplified equation:
[tex]\[ \left(\frac{11}{9}\right)^{-3} = \left(\frac{11}{9}\right)^{2n-1} \][/tex]
Next, set the exponents on both sides of the equation equal to each other:
[tex]\[ -3 = 2n - 1 \][/tex]
Solving for \( n \):
First, add 1 to both sides of the equation:
[tex]\[ -3 + 1 = 2n \][/tex]
[tex]\[ -2 = 2n \][/tex]
Next, divide both sides by 2:
[tex]\[ n = \frac{-2}{2} = -1 \][/tex]
So, the value of \( n \) is \( -1 \).
Now, let's summarize the solution:
1. Simplify the left-hand side of the equation using the properties of exponents: \( \left(\frac{11}{9}\right)^{-3} \).
2. Set the exponents equal: \( -3 = 2n - 1 \).
3. Solve for \( n \): \( n = -1 \).
Thus, the value of [tex]\( n \)[/tex] is [tex]\( -1 \)[/tex], and the exponents on both sides of the equation are [tex]\( -3 \)[/tex].
