Simplify the following: $\left(p^{2+3 n}\right)\left(p^{2 n}\right)$
Answer (1)
Apply the exponent rule a m ⋅ a n = a m + n .
Add the exponents: ( 2 + 3 n ) + ( 2 n ) .
Combine like terms: 2 + 3 n + 2 n = 5 n + 2 .
The simplified expression is p 5 n + 2 .
Explanation
Understanding the Problem We are given the expression ( p 2 + 3 n ) ( p 2 n ) to simplify.
Identifying the Rule The base of both exponential terms is p . We need to use the property of exponents: a m ⋅ a n = a m + n .
Applying the Exponent Rule Applying the rule of exponents a m ⋅ a n = a m + n to the given expression, we add the exponents: ( 2 + 3 n ) + ( 2 n ) .
Simplifying the Exponent Combining like terms in the exponent, we have 2 + 3 n + 2 n = 2 + 5 n = 5 n + 2 .
Final Result Therefore, the simplified expression is p 5 n + 2 .
Examples
Understanding how to simplify expressions with exponents is crucial in various fields, such as physics and engineering. For example, when dealing with wave functions or signal processing, you often encounter exponential terms. Simplifying these terms allows for easier analysis and manipulation of complex equations. Imagine you are analyzing the intensity of a light wave, which is proportional to the square of the amplitude. If the amplitude is given by an exponential function, simplifying the expression helps in determining the intensity accurately.
