Step 1 : Understanding the Question
This algebraic problem involves laws of indices (exponents). The goal is to determine the value of a variable $p$ by solving an exponential equation and then using that value to calculate the expression $2p + 1$. The fundamental requirement for solving such equations is to ensure that the mathematical bases on both sides of the "equals" sign are identical, which allows for the direct comparison of their powers.
Step 2 : Key Formulas and approach
The approach focuses on "Base Equalization." We recognize that 9 and 25 are the squares of 3 and 5, respectively, which allows us to rewrite the entire equation using the common base of $\frac{3}{5}$. We then apply standard exponent rules to simplify the expression.
Key Formulas:
1. $(a^x)^y = a^{xy}$
2. $a^x \cdot a^y = a^{x+y}$
3. If $a^x = a^y$, then $x = y$
Step 3 : Detailed Explanation
Base Transformation: The equation contains $\frac{9}{25}$. We convert this to the base of $\frac{3}{5}$ by noting that $\frac{9}{25} = \left(\frac{3}{5}\right)^2$. Thus, $\left(\frac{9}{25}\right)^8$ becomes $\left[\left(\frac{3}{5}\right)^2\right]^8 = \left(\frac{3}{5}\right)^{16}$.
Simplifying the Right-Hand Side (RHS): Now the RHS of the equation is $\left(\frac{3}{5}\right)^{16} \times \left(\frac{3}{5}\right)^{-3}$. Using the product rule for exponents, we add the powers: $16 + (-3) = 13$. The RHS simplifies to $\left(\frac{3}{5}\right)^{13}$.
Equating the Powers: The full equation is now $\left(\frac{3}{5}\right)^{3p-2} = \left(\frac{3}{5}\right)^{13}$. Since the bases are now the same, we can equate the exponents: $3p - 2 = 13$.
Solving for p: We add 2 to both sides to get $3p = 15$. Dividing by 3 gives us $p = 5$.
Final Expression Calculation: The question asks for the value of $2p + 1$. Substituting $p = 5$, we get $2(5) + 1 = 10 + 1 = 11$.
Step 4 : Final Answer
The value of the expression $2p + 1$ is 11, making (B) the correct option.