To solve the equation \(\frac{9^{2x-1}}{27^{x+1}} = 1\), we will first rewrite the bases in terms of powers of 3, since both 9 and 27 can be expressed as powers of 3. ### Step 1: Rewrite the bases - We know that \(9 = 3^2\) and \(27 = 3^3\). - Therefore, we can rewrite the equation as follows: \[ \frac{(3^2)^{2x-1}}{(3^3)^{x+1}} = 1 \] ### Step 2: Simplify the expression Using the power of a power property \((a^m)^n = a^{m \cdot n}\), we can simplify the numerator and the denominator: \[ \frac{3^{2(2x-1)}}{3^{3(x+1)}} = 1 \] This simplifies to: \[ \frac{3^{4x-2}}{3^{3x+3}} = 1 \] ### Step 3: Apply the quotient rule for exponents Using the quotient rule for exponents, which states that \(\frac{a^m}{a^n} = a^{m-n}\), we can further simplify: \[ 3^{(4x-2) - (3x+3)} = 1 \] This simplifies to: \[ 3^{4x - 2 - 3x - 3} = 1 \] ### Step 4: Combine like terms Now, combine the terms in the exponent: \[ 3^{(4x - 3x) - (2 + 3)} = 3^{x - 5} = 1 \] ### Step 5: Solve for the exponent We know that \(3^0 = 1\). Therefore, we can set the exponent equal to zero: \[ x - 5 = 0 \] ### Step 6: Solve for \(x\) Now, solve for \(x\): \[ x = 5 \] ### Conclusion Thus, the value of \(x\) is \(5\). ### Explanation of Options - **Option A (8)**: This is incorrect because substituting \(x = 8\) into the original equation does not satisfy it. - **Option B (5)**: This is correct as we have shown through our calculations that \(x = 5\) satisfies the equation. - **Option C (3)**: This is incorrect for the same reason as option A; substituting \(x = 3\) does not satisfy the original equation. - **Option D (2)**: This is also incorrect; substituting \(x = 2\) does not satisfy the original equation. ### Revision Summary - Rewrite bases in terms of a common base (in this case, powers of 3). - Use properties of exponents to simplify the equation. - Set the exponent equal to zero when the expression equals 1. - Solve for the variable to find the correct answer. The correct answer is **B. 5**.