Question - Simplification of Exponential Expressions
Solution:
To simplify \(10^{-8}\), we can express it as:
\(10^{-8} = \frac{1}{10^{8}}\)
Tutor Solution:
To simplify \(10^{-8}\), we need to understand the concept of negative exponents.
A negative exponent indicates that we should take the reciprocal of the base raised to the corresponding positive exponent. This is expressed mathematically as:
\[ a^{-n} = \frac{1}{a^n} \]where \(a\) is any non-zero number, and \(n\) is a positive integer.
In our case, we have:
\[ 10^{-8} = \frac{1}{10^8} \]This means we are finding the reciprocal of \(10^8\).
Next, let's calculate \(10^8\). The value \(10^8\) means \(10\) multiplied by itself \(8\) times:
\[ 10^8 = 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 100000000 \]Now substituting this back into our expression gives:
\[ 10^{-8} = \frac{1}{10^8} = \frac{1}{100000000} \]Thus, \(10^{-8}\) simplifies to:
\[ 10^{-8} = 0.00000001 \]To summarize, we simplified \(10^{-8}\) using the property of negative exponents, which resulted in:
\[ 10^{-8} = \frac{1}{100000000} \text{ or } 0.00000001 \]This process shows how negative exponents can be understood as reciprocals, making it easier to simplify such expressions.
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