What is 4 to the power of -3
The Correct Answer and Explanation is :
To calculate (4) to the power of (-3), we can express it mathematically as:
[
4^{-3} = \frac{1}{4^3}
]
Now, we need to calculate (4^3):
[
4^3 = 4 \times 4 \times 4 = 64
]
Substituting back into the equation, we get:
[
4^{-3} = \frac{1}{64}
]
Thus, the answer is:
[
4^{-3} = \frac{1}{64}
]
Explanation
Exponents and Negative Powers
Exponents are a way to represent repeated multiplication. For example, (a^n) means (a) is multiplied by itself (n) times. When the exponent is negative, it indicates the reciprocal of the positive exponent. This concept is rooted in the properties of numbers and how they operate under multiplication and division.
Understanding Negative Exponents
The rule for negative exponents states that for any non-zero number (a), (a^{-n} = \frac{1}{a^n}). This rule allows us to convert negative exponents into positive ones, making calculations easier. In our case, (4^{-3}) was rewritten as (\frac{1}{4^3}).
Calculating (4^3)
Next, we computed (4^3). Multiplying (4) by itself three times gives us (64). This calculation demonstrates how exponents can represent larger numbers compactly.
Final Result
By substituting (4^3) back into our expression for (4^{-3}), we find that (4^{-3} = \frac{1}{64}). Understanding these concepts is crucial in mathematics, especially in algebra and calculus, where exponents frequently appear.
In summary, (4^{-3}) simplifies to (\frac{1}{64}), illustrating the powerful relationship between positive and negative exponents and their practical implications in mathematics.