Simplify each expression. See Example 1.9³ • 9⁵

Verified step by step guidance

1. Identify the base and the exponents in the expression: the base is 9, and the exponents are 3 and 5.
2. Recall the property of exponents that states: \( a^m \cdot a^n = a^{m+n} \).
3. Apply this property to the expression: \( 9^3 \cdot 9^5 = 9^{3+5} \).
4. Simplify the exponent by adding the numbers: \( 3 + 5 \).
5. Rewrite the expression with the simplified exponent: \( 9^8 \).

Key Concepts

Exponential Rules

Exponential rules are fundamental properties that govern the manipulation of expressions involving exponents. One key rule states that when multiplying two expressions with the same base, you can add their exponents. For example, a^m • a^n = a^(m+n). This principle is essential for simplifying expressions like 9³ • 9⁵.

Base of an Exponent

The base of an exponent is the number that is being raised to a power. In the expression 9³, the base is 9, and it is raised to the power of 3. Understanding the base is crucial because it determines the value of the expression when combined with the exponent, especially when applying exponential rules.

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 exponents, this often means applying the rules of exponents to combine terms. For instance, simplifying 9³ • 9⁵ results in 9^(3+5) = 9⁸, showcasing the process of simplification.