Problem 27 - Elementary Algebra

Question

Use the product rule and quotient rule of exponents to simplify the following problems. Assume that all bases are nonzero and that all exponents are whole numbers. $$ 3^{2} \cdot 3^{3} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(3^{2} \cdot 3^{3}\) using the product rule of exponents. Answer: \(3^{5}\)

Step 1: Recall the product rule of exponents

The product rule of exponents states that when multiplying two powers with the same base, the exponents are added together. Mathematically, it can be written as: $$ a^{m} \cdot a^{n} = a^{m + n} $$

Step 2: Apply the product rule to the given expression

We can now apply the product rule to the given expression: $$ 3^{2} \cdot 3^{3} $$ As the bases are the same (3), we can add the exponents: $$ 3^{2} \cdot 3^{3} = 3^{(2 + 3)} $$

Step 3: Calculate the result

Now, we need to calculate the final result by simply adding the exponents: $$ 3^{(2 + 3)} = 3^{5} $$ So, the simplified expression is \(3^{5}\).

Key Concepts

Exponent Rules

One of the fundamental concepts in algebra is understanding exponent rules. These rules are crucial for simplifying expressions that involve powers of numbers. Particularly, the product rule of exponents is a key tool that helps in combining powers with the same base. This operation is fairly straightforward; when you multiply two exponential expressions with the same base, you can keep the base and add the exponents:

• \( a^m \cdot a^n = a^{m+n} \)
• \( a^m / a^n = a^{m-n} \) when \(n \leq m \)
• \( (a^m)^n = a^{m \cdot n} \)
• \( a^0 = 1 \) (as long as \( a \eq 0 \))
• \( a^{-n} = 1 / a^n \)

Each rule is an essential piece of the exponent puzzle, allowing you to manipulate and simplify even the most complex expressions. Remember, these rules are applicable only when the bases of the exponents are the same.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form while retaining their original value. In the realm of exponents, simplification often uses the aforementioned exponent rules to combine and reduce terms. The aim is to minimize the number of exponential terms by combining like bases and to present the expression in the most condensed form possible.

Let's take a practical look at this concept by examining the provided problem. We start with the expression \(3^2 \cdot 3^3\). By applying the product rule, we consolidate it into \(3^{2+3}\), which simplifies further to \(3^5\). This methodical approach to simplification helps in understanding complex problems and finding solutions with greater ease.

Algebraic Operations

Algebraic operations include basic processes such as addition, subtraction, multiplication, and division, yet they extend to more complex operations that involve variables, exponents, and sometimes, imaginary numbers. When doing algebra, it's essential to follow the order of operations—often abbreviated as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)—to ensure accurate results.

Within this hierarchy, operations involving exponents must be addressed after any calculations inside parentheses but before any multiplication or division tasks. Proper application of algebraic operations, especially when combined with other rules like the product rule of exponents, leads to the efficient simplification of algebraic expressions.

Base and Exponent

The two parts of an exponential expression are the base and the exponent. The base is the number that is being multiplied by itself, and the exponent tells us how many times the base is used as a factor. For example, in \(3^5\), 3 is the base, and 5 is the exponent, indicating that 3 is multiplied by itself a total of 5 times.

In algebraic terms, understanding the relationship between the base and the exponent is key to utilizing the rules of exponents effectively. They allow us to perform operations on exponential terms systematically. It is important to acknowledge that different bases are not combined using exponent rules—such operations are subject to different algebraic principles.