3 to the power of -2 as a fraction
The Correct Answer and Explanation is :
To calculate (3^{-2}) and express it as a fraction, we follow the rules of exponents. The negative exponent indicates that we are dealing with the reciprocal of the base raised to the positive exponent.
Step 1: Understand the Negative Exponent
The rule for negative exponents states that:
[
a^{-n} = \frac{1}{a^n}
]
where (a) is any non-zero number and (n) is a positive integer.
Step 2: Apply the Rule to (3^{-2})
Using this rule for (3^{-2}), we can rewrite it as:
[
3^{-2} = \frac{1}{3^2}
]
Step 3: Calculate (3^2)
Next, we need to calculate (3^2):
[
3^2 = 3 \times 3 = 9
]
Step 4: Substitute Back into the Fraction
Now we can substitute (9) back into our fraction:
[
3^{-2} = \frac{1}{9}
]
Thus, the value of (3^{-2}) as a fraction is (\frac{1}{9}).
Explanation of the Result
The expression (3^{-2}) represents the concept of division and reciprocation in mathematics. When we see a negative exponent, it suggests that the number should be moved from the numerator to the denominator (or vice versa) to create a positive exponent. This is a crucial concept in algebra, as it simplifies many calculations and makes understanding fractions easier.
In real-world applications, negative exponents often appear in scientific notation and formulas involving decay or growth rates, emphasizing the importance of being comfortable with this concept. By mastering the manipulation of exponents, including those with negative values, you can tackle more complex problems across various mathematical disciplines with confidence.