- Identify the base \(x\) and the exponents \(3\) and \(5\).
- Use the rule for multiplying powers with the same base: \(x^a \cdot x^b = x^{a+b}\).
- Sum the exponents: \(3 + 5 = 8\).
- State the final product: \(\boxed{x^8}\).
Explanation
- Analysis of the problem
We are asked to compute the product of two expressions, \(x^3\) and \(x^5\). Here, the base of both expressions is \(x\), and the exponents are \(3\) and \(5\), respectively. This is a problem involving the rules of exponents.
- Applying the exponent rule
To multiply powers that have the same base, we use the power rule for multiplication:
\[x^a \cdot x^b = x^{a+b}\]
In our case, the base \(x\) is common to both expressions, so we can directly add the exponents \(3\) and \(5\).
- Simplification and final result
Applying the formula \(x^3 \cdot x^5 = x^{3+5}\), we calculate the sum of the exponents: \(3 + 5 = 8\). Thus, the expression simplifies to \(x^8\).
Examples
Knowing how to handle exponents is crucial for fields like computer science and engineering. For example, if you are working with data storage where files grow by powers of two, such as transferring a file of size \(2^3\) gigabytes multiplied by a transmission rate of \(2^5\) times, the exponent rule helps calculate the total size quickly: \(2^3 \times 2^5 = 2^{3+5} = 2^8 = 256\). Mastering these basic rules allows you to manage and scale data efficiently.