Simplify each exponential expression in Exercises 23–64. x^3⋅x^7

Verified step by step guidance

1. Identify the base of the exponential expressions. In this case, both terms have the base \(x\).
2. Recall the property of exponents: when multiplying like bases, you add the exponents. This is expressed as \(a^m \cdot a^n = a^{m+n}\).
3. Apply the property to the given expression: \(x^3 \cdot x^7\).
4. Add the exponents: \(3 + 7\).
5. Rewrite the expression with the new exponent: \(x^{3+7}\).

Key Concepts

Exponential Rules

Exponential rules are fundamental properties that govern the manipulation of exponential expressions. One key rule states that when multiplying two expressions with the same base, you add their exponents. For example, x^a ⋅ x^b = x^(a+b). This rule is essential for simplifying expressions like x^3 ⋅ x^7.

Base of an Exponent

The base of an exponent is the number that is being raised to a power. In the expression x^3, 'x' is the base, and '3' is the exponent. Understanding the role of the base is crucial when applying exponential rules, as operations are only valid when the bases are the same, allowing for simplification through addition of exponents.

Simplification of Expressions

Simplification of expressions involves reducing an expression to its simplest form, making it easier to understand or compute. In the context of exponential expressions, this often means applying the rules of exponents to combine terms. For instance, simplifying x^3 ⋅ x^7 results in x^(3+7) = x^10, demonstrating the process of simplification.