Simplify the expression: [tex]$\left(a^3\right)^2 \cdot a^5$[/tex]
Answer (1)
Apply the power of a power rule: ( a 3 ) 2 = a 3 × 2 = a 6 .
Apply the product of powers rule: a 6 ⋅ a 5 = a 6 + 5 = a 11 .
The simplified expression is a 11 .
Explanation
Understanding the Problem We are asked to simplify the expression ( a 3 ) 2 ⋅ a 5 . To do this, we will use the rules of exponents.
Applying the Power of a Power Rule First, we simplify the term ( a 3 ) 2 . According to the power of a power rule, when we raise a power to a power, we multiply the exponents: ( a m ) n = a m ⋅ n . In this case, we have ( a 3 ) 2 = a 3 ⋅ 2 = a 6 .
Applying the Product of Powers Rule Now we have a 6 ⋅ a 5 . According to the product of powers rule, when we multiply powers with the same base, we add the exponents: a m ⋅ a n = a m + n . In this case, we have a 6 ⋅ a 5 = a 6 + 5 = a 11 .
Final Answer Therefore, the simplified expression is a 11 .
Examples
Understanding how to simplify expressions with exponents is crucial in many areas, such as calculating compound interest or analyzing exponential growth in populations. For example, if a population of bacteria doubles every hour, the growth can be modeled using exponents. Simplifying these exponential expressions helps predict the population size after a certain number of hours. Similarly, in finance, compound interest calculations involve exponents, and simplifying these expressions can help determine the future value of an investment.
