Match each product of powers with its simplified expression.
| | | |
| :----------------- | :-: | :------ |
| [tex]$5^{-4} \cdot 5^4 \cdot 5^0$[/tex] | | [tex]$\frac{1}{5^6}$[/tex] |
| [tex]$5^{-3} \cdot 5^{-3}$[/tex] | | [tex]$5^4$[/tex] |
| [tex]$5^6 \cdot 5^{-4}$[/tex] | | [tex]$5^{10}$[/tex] |
| [tex]$5^7 \cdot 5^3$[/tex] | | [tex]$5^2$[/tex] |
| [tex]$5 \cdot 5^3$[/tex] | | [tex]$5^0$[/tex] |
Answer (1)
Simplify each expression using the rule a m ". a n = a m + n .
5 − 4 ". 5 4 ". 5 0 = 5 0
5 − 3 ". 5 − 3 = 5 6 1
5 6 ". 5 − 4 = 5 2
5 7 ". 5 3 = 5 10
5". 5 3 = 5 4
Match the simplified expressions to the options provided.
Explanation
Understanding the Problem We are given 5 expressions involving products of powers of 5 and 5 simplified expressions. We need to match each product of powers with its simplified expression. The expressions involve negative and positive exponents. We will use the rule a m ". a n = a m + n to simplify the expressions.
Solution Plan We will simplify each product of powers using the rule a m ". a n = a m + n .
Simplifying the first expression
5 − 4 ". 5 4 ". 5 0 = 5 − 4 + 4 + 0 = 5 0
Simplifying the second expression
5 − 3 ". 5 − 3 = 5 − 3 + ( − 3 ) = 5 − 6 = 5 6 1
Simplifying the third expression
5 6 ". 5 − 4 = 5 6 + ( − 4 ) = 5 2
Simplifying the fourth expression
5 7 ". 5 3 = 5 7 + 3 = 5 10
Simplifying the fifth expression
5". 5 3 = 5 1 ". 5 3 = 5 1 + 3 = 5 4
Matching the expressions Now, we match the simplified expressions with the given options.
Examples
Understanding exponential rules is crucial in many fields, such as calculating compound interest or modeling population growth. For example, if a population doubles every 10 years, we can use exponential growth to predict its size in the future. Similarly, in finance, understanding how interest accumulates over time involves exponential calculations, allowing investors to estimate their returns accurately. These concepts are not just theoretical; they have practical applications in everyday decision-making and long-term planning.
