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Chapter 4: Problem 1

Find each product or quotient. Express using exponents. $$9^{3} \cdot 9^{2}$$

Short Answer

Expert verified
The product is \(9^5\).

Step by step solution

01

Identify the Base

Both terms in the expression, \(9^3\) and \(9^2\), have the same base, which is 9. This is important because it allows us to combine the terms using exponential rules.
02

Apply Product of Powers Rule

When multiplying expressions with the same base, we add the exponents. Therefore, \(9^3 \cdot 9^2 = 9^{3+2}\).
03

Simplify the Exponent

Add the exponents together: \(3+2=5\). Thus, the expression simplifies to \(9^5\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Product of Powers Rule
The product of powers rule is a basic principle in mathematics that helps us deal with exponents when multiplying terms with the same base. This rule is quite simple: if you have two powers with the same base, you add their exponents. The general formula is:\[a^m \cdot a^n = a^{m+n}\]where \(a\) is the base and \(m\) and \(n\) are the exponents. So, when you multiply terms that have the same base, you don't multiply the bases, but rather, you work with their exponents.
  • Think of it as counting all the times a number is being multiplied by itself.
  • This rule simplifies the process, avoiding the need for large and cumbersome calculations.
You'll often encounter this in algebra, helping streamline calculations involving exponents.
Base Identification
Identifying the base is the first critical step in applying the product of powers rule. The base of an expression provides the groundwork upon which exponents operate. In the expression \(9^3 \cdot 9^2\), the base is 9 for both terms. Spotting this correctly is crucial because only terms with the same base can use the product of powers rule.
  • Check both terms or expressions carefully to ensure the base matches before proceeding.
  • If the base is different, you cannot use the product of powers rule directly.
This foundational understanding will aid you immensely when simplifying exponential expressions.
Multiplying Expressions with the Same Base
Multiplying expressions with the same base means that you can combine these expressions using the product of powers rule. Once you confirm that the bases are the same, you can proceed to add the exponents.To illustrate, in the expression \(9^3 \cdot 9^2\): - You identify the base as 9. - Then, you add the exponents: 3 and 2, resulting in 5.- Therefore, the expression becomes \(9^5\).
  • This method is efficient and avoids expanding the terms, which can be cumbersome for large exponents.
  • It's especially useful in algebra and higher-level math where such operations occur frequently.
By mastering this, you simplify problems and make calculations more straightforward.

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