Write the expression as an exponent:
[tex]$3^4 \cdot 3^2 \cdot 3^3 \cdot 3$[/tex]
Answer (1)
We have the expression 3 4 c d o t 3 2 c d o t 3 3 c d o t 3 .
Apply the rule of exponents: a m c d o t a n = a m + n .
Add the exponents: 4 + 2 + 3 + 1 = 10 .
Write the final expression as 3 10 , so the answer is 3 10 .
Explanation
Understanding the problem We are given the expression 3 4 c d o t 3 2 c d o t 3 3 c d o t 3 and we want to write it as an exponent.
Recalling the exponent rule To simplify this expression, we need to remember the rule for multiplying exponents with the same base. The rule states that when you multiply exponential expressions with the same base, you add the exponents. In other words, a m c d o t a n = a m + n .
Applying the exponent rule First, we can rewrite the expression to explicitly show the exponent of the last term: 3 4 c d o t 3 2 c d o t 3 3 c d o t 3 1 . Now, we add the exponents: 4 + 2 + 3 + 1 = 10 .
Writing the final expression Therefore, the expression 3 4 c d o t 3 2 c d o t 3 3 c d o t 3 can be written as 3 10 .
Examples
Exponents are used to calculate compound interest. For example, if you invest 1000 a t anann u a l in t eres t r a t eo f 5 A = P(1 + r)^t , w h ere A i s t h e am o u n t a f t er t ye a rs , P i s t h e p r in c i p a l am o u n t , an d r i s t h e ann u a l in t eres t r a t e . I n t hi sc a se , A = 1000(1 + 0.05)^{10} approx 1628.89$. Understanding exponents helps you calculate how your investments grow over time.
