What expression is equivalent to 4 to the negative 3 power
The correct Answer and Explanation is:
The expression (4^{-3}) is equivalent to (\frac{1}{4^3}).
Explanation:
When you have an exponent that is a negative number, such as (-3), it means you are taking the reciprocal of the base (which is (4) in this case) raised to the positive version of that exponent. So, (4^{-3}) becomes:
[
4^{-3} = \frac{1}{4^3}
]
Now, let’s break down the steps:
- Negative Exponent Rule: This rule states that for any non-zero number (a) raised to a negative exponent (-n), the expression can be rewritten as the reciprocal of the base raised to the positive exponent (n). That is:
[
a^{-n} = \frac{1}{a^n}
]
In our case, (a = 4) and (n = 3), so we rewrite (4^{-3}) as:
[
4^{-3} = \frac{1}{4^3}
] - Calculate (4^3): Now, calculate the value of (4^3), which means multiplying (4) by itself three times:
[
4^3 = 4 \times 4 \times 4 = 64
] - Final Expression: After calculating (4^3), the expression becomes:
[
\frac{1}{4^3} = \frac{1}{64}
]
Thus, the expression (4^{-3}) simplifies to (\frac{1}{64}).
Why This Works:
Negative exponents indicate division rather than multiplication. By using the reciprocal (flipping the base), we avoid using negative values in the calculation. This concept is essential in algebra and higher-level mathematics, allowing us to work with inverse powers of numbers. For example, (x^{-1} = \frac{1}{x}) demonstrates how negative exponents transform expressions into fractions.
So, the correct answer to the expression (4^{-3}) is (\frac{1}{64}).