Equation Transformation Using Exponent Rules

We are given the exponential equation: $ \left(\frac{7}{11}\right)^{k-5} = \left(\frac{11}{7}\right)^{k-9} $ The key to solving this equation is to make the bases on both sides the same. We know that the reciprocal of a number can be expressed using a negative exponent. Notice that $\frac{11}{7}$ is the reciprocal of $\frac{7}{11}$. Therefore, we can rewrite $\frac{11}{7}$ as $\left(\frac{7}{11}\right)^{-1}$.

Let's apply this to the right side of the equation:

Solving for the Variable $k$

Now that we have the same base, $\frac{7}{11}$, on both sides of the equation, we can equate the exponents. This is based on the property that if $a^x = a^y$ and $a \neq 0, 1, -1$, then $x=y$.

Equating the exponents:

To solve for $k$, we first gather the $k$ terms on one side and the constants on the other.

• Add $k$ to both sides: $ k + k - 5 = 9 - k + k $ $ 2k - 5 = 9 $
• Add 5 to both sides: $ 2k - 5 + 5 = 9 + 5 $ $ 2k = 14 $
• Divide by 2: $ \frac{2k}{2} = \frac{14}{2} $ $ k = 7 $

We have found the value of $k$ to be $7$. The question asks for the value of $2^k$.

Calculating the Value of $2^k$

Now we substitute the value of $k$ we found ($k=7$) into the expression $2^k$.

Calculating $2^7$: $ 2^7 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 $ $ 2^7 = 4 \times 2 \times 2 \times 2 \times 2 \times 2 $ $ 2^7 = 8 \times 2 \times 2 \times 2 \times 2 $ $ 2^7 = 16 \times 2 \times 2 \times 2 $ $ 2^7 = 32 \times 2 \times 2 $ $ 2^7 = 64 \times 2 $ $ 2^7 = 128 $

Therefore, the value of $2^k$ is $128$.