What is 3 to the power of -4 as a fraction?
The Correct Answer and Explanation is :
To express ( 3^{-4} ) as a fraction, we first need to understand what a negative exponent means. In mathematics, a negative exponent indicates the reciprocal of the base raised to the positive exponent. The general rule for negative exponents is:
[
a^{-n} = \frac{1}{a^n}
]
where ( a ) is any non-zero number and ( n ) is a positive integer. Applying this rule to ( 3^{-4} ):
[
3^{-4} = \frac{1}{3^4}
]
Next, we calculate ( 3^4 ). This means multiplying 3 by itself four times:
[
3^4 = 3 \times 3 \times 3 \times 3
]
Calculating step-by-step:
- ( 3 \times 3 = 9 )
- ( 9 \times 3 = 27 )
- ( 27 \times 3 = 81 )
Thus, ( 3^4 = 81 ). Now, substituting this back into the equation gives us:
[
3^{-4} = \frac{1}{3^4} = \frac{1}{81}
]
So, ( 3^{-4} ) as a fraction is ( \frac{1}{81} ).
Explanation of Negative Exponents
Negative exponents may seem confusing at first, but they are simply another way to express division. When we see a negative exponent, we should think of it as the inverse of the base raised to the positive exponent. This helps to maintain consistency in mathematical operations and allows for simpler manipulation of expressions.
Understanding negative exponents is crucial in algebra and higher-level mathematics, as it lays the groundwork for working with fractions, rational expressions, and functions. When we encounter expressions with negative exponents, converting them to their positive counterparts ensures clarity and ease of calculation.
In summary, the value of ( 3^{-4} ) expressed as a fraction is ( \frac{1}{81} ). This transformation demonstrates the fundamental properties of exponents and their practical applications in mathematics.