Why is 2 to the power of -1 equal to 0.5 or 1/2
The Correct Answer and Explanation is :
The correct answer is: 0.5
When we raise a number to a negative exponent, it signifies taking the reciprocal of the base raised to the positive version of that exponent. To explain why ( 2^{-1} ) equals 0.5 or ( \frac{1}{2} ), let’s break this down step by step:
Basic Exponent Rules
For any base ( a ), raising it to a negative exponent ( n ) follows this rule:
[
a^{-n} = \frac{1}{a^n}
]
This means that instead of multiplying the base by itself ( n ) times, we take the reciprocal of ( a ) raised to the positive ( n ).
Application to 2 Raised to the Power of -1
Using this rule:
[
2^{-1} = \frac{1}{2^1}
]
Since ( 2^1 = 2 ), we get:
[
2^{-1} = \frac{1}{2}
]
Which equals 0.5 as a decimal.
Why Does This Work?
To understand this further, consider how exponents behave in general:
- Positive exponents: ( 2^3 ) means multiplying 2 by itself three times: ( 2 \times 2 \times 2 = 8 ).
- Zero exponent: ( 2^0 = 1 ), because any non-zero number raised to the power of 0 equals 1.
- Negative exponents: Negative exponents reverse the process. Instead of repeated multiplication, negative exponents indicate repeated division. So ( 2^{-1} ) means dividing 1 by 2 once, which gives us ( \frac{1}{2} ) or 0.5.
Conclusion
The expression ( 2^{-1} ) means we are asking for the reciprocal of ( 2 ), which is ( \frac{1}{2} ), or 0.5 in decimal form. The concept of negative exponents is a natural extension of the rules of exponents, allowing us to easily handle inverses of numbers raised to powers.