Simplify the following.
[tex]7^1=\square[/tex]
[tex]4^0=\square[/tex]
[tex]6^{-1}=\square[/tex]
Answer (1)
Any number raised to the power of 1 is itself: 7 1 = 7 .
Any non-zero number raised to the power of 0 is 1: 4 0 = 1 .
A number raised to the power of -1 is its reciprocal: 6 − 1 = 6 1 .
Therefore, the simplified expressions are 7 , 1 , 6 1 .
Explanation
Understanding the Problem We are asked to simplify three expressions involving exponents: 7 1 , 4 0 , and 6 − 1 . Let's tackle each one individually.
Simplifying 7^1 Any number raised to the power of 1 is simply the number itself. Therefore, 7 1 = 7 .
Simplifying 4^0 Any non-zero number raised to the power of 0 is equal to 1. Therefore, 4 0 = 1 .
Simplifying 6^{-1} A number raised to the power of -1 is equal to its reciprocal, which means a − 1 = a 1 . Thus, 6 − 1 = 6 1 .
Final Answer In summary, we have 7 1 = 7 , 4 0 = 1 , and 6 − 1 = 6 1 .
Examples
Exponents are used everywhere in science and engineering. For example, when calculating the area of a square with side length 7, we compute 7 2 = 49 . When dealing with very small quantities, like the size of an atom, we use negative exponents. If something grows exponentially, like the number of bacteria in a petri dish, we can use exponents to model that growth. Understanding exponents is crucial for understanding many real-world phenomena.
