\( 4 \times \) An equation is shown. \( \left(3^{4}\right)^{p}=3^{32} \) What is the value of \( p ? \) \( 4 \times \quad \) A \( \quad \) B \( 4 \times 8 \) \( 4 \times 8 \)

To solve the equation \( (3^{4})^{p} = 3^{32} \), we can use the power of a power property: \( (a^{m})^{n} = a^{m \cdot n} \). Thus, we have \( 3^{4p} = 3^{32} \). Since the bases are the same, we can set the exponents equal to each other: \( 4p = 32 \). Now, divide both sides by 4 to find \( p \): \[ p = \frac{32}{4} = 8. \] So, the value of \( p \) is 8! In the realm of exponential equations, they play vital roles in various fields such as finance (with compound interest calculations) and physics (like radioactive decay models). Understanding how to manipulate exponents can provide insights into complex growth and decay patterns, making it easier to model real-world phenomena. On the academic side, don’t forget to explore how exponent laws are applicable in simplifying expressions encountered in algebra and calculus. The more you practice, the more you will uncover the fun and intriguing relationships within mathematics!