7 to the power of 4 times 7 to the power of negative power of 4
The Correct Answer and Explanation is:
To solve the expression ( 7^4 \times 7^{-4} ), we can use the properties of exponents. Specifically, when multiplying two terms with the same base, we apply the product of powers rule, which states:
[
a^m \times a^n = a^{m+n}
]
Where:
- ( a ) is the base (in this case, 7),
- ( m ) and ( n ) are the exponents.
Step 1: Apply the product of powers rule
Given the expression ( 7^4 \times 7^{-4} ), we can apply the product of powers rule:
[
7^4 \times 7^{-4} = 7^{4 + (-4)} = 7^{0}
]
Step 2: Simplify the exponent
When any non-zero number is raised to the power of 0, the result is always 1:
[
7^0 = 1
]
Step 3: Final Answer
Thus, the value of ( 7^4 \times 7^{-4} ) simplifies to:
[
\boxed{1}
]
Explanation of the Concepts Involved
The operation relies on understanding the rules of exponents. Exponents indicate how many times a number (the base) is multiplied by itself. In the case of positive exponents, such as ( 7^4 ), the base is multiplied four times:
[
7^4 = 7 \times 7 \times 7 \times 7 = 2401
]
However, when dealing with negative exponents, such as ( 7^{-4} ), the base is divided rather than multiplied. The negative exponent tells us to take the reciprocal of the base raised to the positive exponent:
[
7^{-4} = \frac{1}{7^4} = \frac{1}{2401}
]
Thus, multiplying ( 7^4 ) and ( 7^{-4} ) involves multiplying 2401 by its reciprocal, which equals 1:
[
2401 \times \frac{1}{2401} = 1
]
This is why ( 7^4 \times 7^{-4} = 7^0 = 1 ).