Write the expression as an exponent: [tex]$2^9 \cdot 32$[/tex]
Answer (2)
Rewrite 32 as a power of 2: 32 = 2 5 .
Apply the exponent property $a^m
\cdot a^n = a^{m+n}$.
Calculate the sum of the exponents: 9 + 5 = 14 .
Write the final expression as an exponent: 2 14 .
Explanation
Understanding the problem We are asked to write the expression $2^9
\cdot 32$ as an exponent. This means we want to express the given product as a single power.
Rewriting 32 as a power of 2 First, we need to rewrite 32 as a power of 2. We know that 32 = 2 5 . So, we can rewrite the expression as:
2 9 ⋅ 2 5
Using the exponent property Now, we use the property of exponents that says when you multiply two powers with the same base, you add the exponents: $a^m
\cdot a^n = a^{m+n} . I n o u rc a se , a=2 , m=9 , an d n=5$. Therefore,
2 9 ⋅ 2 5 = 2 9 + 5 = 2 14
Final Answer So, the expression $2^9
\cdot 32$ can be written as 2 14 .
Examples
Exponents are used to describe exponential growth or decay in various real-world scenarios. For example, the growth of a bacteria colony can be modeled using exponents. If a bacteria colony doubles every hour, the number of bacteria after t hours can be represented as $N = N_0
\cdot 2^t , w h ere N_0 i s t h e ini t ia l n u mb ero f ba c t er ia . S imi l a r l y , co m p o u n d in t eres t c a l c u l a t i o n s u see x p o n e n t s t o d e t er min e t h e f u t u re v a l u eo f anin v es t m e n t . I f yo u in v es t P d o ll a rs a t anann u a l in t eres t r a t eo f r co m p o u n d e d n t im es p erye a r , t h e am o u n t A yo u ′ ll ha v e a f t er t ye a rs i s A = P(1 + \frac{r}{n})^{nt}$. Understanding exponents is crucial for modeling and analyzing these types of phenomena.
The expression 2 9 ⋅ 32 can be rewritten as an exponent by first expressing 32 as 2 5 . Then, using the property of exponents that allows us to add exponents when multiplying like bases, we combine them to get 2 14 . Thus, the final answer is 2 14 .
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