- Identify the division property of exponents: \(\frac{x^m}{x^n} = x^{m-n}\).
- Substitute the given exponents: \(m = 3\) and \(n = -5\).
- Simplify the resulting exponent: \(3 - (-5) = 3 + 5 = 8\).
- The final exponent is \(\boxed{8}\).
Explanation
- Understanding the Exponent Laws
The problem requires us to simplify the expression \(\frac{x^3}{x^{-5}}\) and express it in the form \(x^n\), where \(n\) is an integer. Recall the fundamental laws of exponents which state that when dividing two powers with the same base, we subtract the exponent of the denominator from the exponent of the numerator: \(\frac{x^m}{x^n} = x^{m-n}\). In our specific expression, the exponent in the numerator is \(m = 3\) and the exponent in the denominator is \(n = -5\).
- Simplifying the Expression
Applying the division rule \(\frac{x^m}{x^n} = x^{m-n}\) to our expression, we substitute \(m = 3\) and \(n = -5\) into the exponent:
\[\frac{x^3}{x^{-5}} = x^{3 - (-5)}\]
Since subtracting a negative number is equivalent to adding its positive counterpart, we simplify the exponent:
\[3 - (-5) = 3 + 5 = 8\]
Therefore, \(\frac{x^3}{x^{-5}} = x^8\). The value that replaces the question mark is 8.
Examples
This concept of exponent rules is essential in fields like computer science and physics, where numbers are often written in scientific notation to represent very large or very small values. For instance, if you are working with data storage units, dividing \(10^{12}\) bytes (terabytes) by \(10^6\) bytes (megabytes) effectively uses this exponent subtraction rule: \(10^{12} / 10^6 = 10^{12-6} = 10^6\), showing that a terabyte is \(10^6\) megabytes. Understanding these rules allows for efficient calculation and simplification of complex expressions.