Question

If

\[ \left(\frac{3}{5}\right)^{3p-2} = \left(\frac{9}{25}\right)^8 \times \left(\frac{3}{5}\right)^{-3} \] 

then the value of \( 2p + 1 \) is:

• A. 10
• B. 11
• C. 14
• D. 15

Hint:

Always simplify the bases to the smallest prime-based fraction possible. Here, seeing that 9 and 25 are squares of 3 and 5 immediately points to the common base.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
To solve equations involving exponents, we must express all terms with the same base. Once the bases are identical, we can equate the exponents.

Step 2: Key Formula or Approach:
The properties of exponents used are: 1. $(a^m)^n = a^{m \times n}$ 2. $a^m \times a^n = a^{m+n}$ 3. If $a^x = a^y$, then $x = y$.

Step 3: Detailed Explanation:
Given: \[ \left(\frac{3}{5}\right)^{3p-2} = \left(\frac{9}{25}\right)^8 \times \left(\frac{3}{5}\right)^{-3} \] Step A: Convert $\frac{9}{25}$ to base $\frac{3}{5}$. Since $\frac{9}{25} = \left(\frac{3}{5}\right)^2$, the equation becomes: \[ \left(\frac{3}{5}\right)^{3p-2} = \left[\left(\frac{3}{5}\right)^2\right]^8 \times \left(\frac{3}{5}\right)^{-3} \] Step B: Simplify the right side. \[ \left(\frac{3}{5}\right)^{3p-2} = \left(\frac{3}{5}\right)^{16} \times \left(\frac{3}{5}\right)^{-3} \] \[ \left(\frac{3}{5}\right)^{3p-2} = \left(\frac{3}{5}\right)^{16 - 3} \] \[ \left(\frac{3}{5}\right)^{3p-2} = \left(\frac{3}{5}\right)^{13} \] Step C: Equate the exponents. \[ 3p - 2 = 13 \] \[ 3p = 15 \implies p = 5 \] Step D: Calculate $2p + 1$. \[ 2(5) + 1 = 11 \]

Step 4: Final Answer:
The value of $2p + 1$ is 11.