Evaluate the expression. (7 + 3)^0
The Correct Answer and Explanation is :
The correct answer is: 1
To evaluate the expression ((7 + 3)^0), we first simplify the expression inside the parentheses:
[
7 + 3 = 10
]
Now, we have:
[
(7 + 3)^0 = 10^0
]
According to the properties of exponents, any non-zero number raised to the power of zero is equal to 1. This can be expressed as:
[
a^0 = 1 \quad \text{for any } a \neq 0
]
In our case, since 10 is a non-zero number, we apply this rule:
[
10^0 = 1
]
Thus, the final answer to the expression ((7 + 3)^0) is:
[
\boxed{1}
]
Explanation of the Concept
The rule that any non-zero number raised to the power of zero equals one can be understood through a few mathematical principles.
- Exponent Rules: Exponents represent repeated multiplication. For example, (a^3 = a \times a \times a). As you decrease the exponent by one, the base remains but you divide by that base. For instance, (a^3/a^1 = a^{3-1} = a^2). If you continue this process, when you reach (a^1/a^1), you get (a^0), which simplifies to 1, assuming (a) is not zero.
- Patterns in Powers: If we look at the pattern of powers of a number, such as (2), we can observe:
- (2^3 = 8)
- (2^2 = 4)
- (2^1 = 2)
- (2^0 = 1)
- (2^{-1} = \frac{1}{2}) Notice how as we decrease the exponent by 1, we keep dividing by 2. Eventually, when we reach (2^0), it must equal 1 to maintain the consistency of the division by 2.
- Applications: Understanding this property is essential in algebra and calculus, as it helps simplify expressions and solve equations involving exponents.
In conclusion, the evaluation of ((7 + 3)^0) leads us to the answer 1, highlighting the significance of the exponent rule across various mathematical contexts.