10 to the power of -2 as a fraction
The Correct Answer and Explanation is:
To express (10^{-2}) as a fraction, we first need to understand what the notation (10^{-2}) means.
Step 1: Understanding Negative Exponents
A negative exponent indicates that we need to take the reciprocal of the base and then raise it to the positive value of the exponent. So, (10^{-2}) is equivalent to:
[
10^{-2} = \frac{1}{10^2}
]
Step 2: Simplifying the Expression
Now, we simplify (10^2). Since (10^2 = 10 \times 10 = 100), we can substitute this value into the reciprocal expression:
[
\frac{1}{10^2} = \frac{1}{100}
]
Thus, (10^{-2}) is equal to (\frac{1}{100}).
Step 3: Verifying the Calculation
We can check our work by converting (\frac{1}{100}) back into its decimal form. The fraction (\frac{1}{100}) is equal to 0.01, and since (10^{-2} = 0.01), our conversion is correct.
Step 4: Explanation of the Process
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. In this case, (a = 10) and (n = 2), so applying this rule gives us:
[
10^{-2} = \frac{1}{10^2} = \frac{1}{100}
]
This transformation is helpful in various mathematical contexts, especially when working with very large or very small numbers. By converting negative exponents into fractions, we can express numbers in more manageable forms, particularly when dealing with scientific notation or precision calculations.
Conclusion
In conclusion, (10^{-2}) as a fraction is (\frac{1}{100}). This conversion is based on the rule for negative exponents, which indicates taking the reciprocal of the base raised to the positive exponent.