Problem 95 - Elementary Algebra Within Reach

Question

Simplify the expression. $$5 z^{3}\left(z^{2}\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(5 z^{5}\).

Step 1: Identify Operation Type

Here you can see that there is a multiplication operation for similar bases. According to the rules of exponents, in case of multiplication with similar bases, the exponents are added together.

Step 2: Add the exponents

So, the expression becomes \(5z^{(3+2)}\) which can be simplified to \(5 z^{5}\).

Step 3: Write down the final answer

The final simplified expression is therefore \(5 z^{5}\).

Key Concepts

Multiplication of Exponents

When you multiply expressions with exponents, it’s crucial to understand how exponents behave. This involves recognizing when two expressions have the same base. If they do, we can simplify them by adding their exponents together. This principle makes it much easier to handle complex expressions.
Let's look at an example to make it clearer:

• Suppose you have the expression \( z^3 \times z^2 \).
• Both terms have the same base \( z \), so you keep the base as it is.
• Then, simply add the exponents: \( 3 + 2 = 5 \).
• The simplified expression is \( z^5 \).

In essence, when multiplying like bases, just add their exponents! This makes handling problems more efficient and less daunting.

Identifying Operation Types in Algebra

Before diving into the arithmetic of any expression in algebra, it's important to first identify the operations involved. This understanding helps you decide which algebraic rules to apply. In algebra, the most common operations include addition, subtraction, multiplication, and division.
For our example problem with \( 5 z^3 (z^2) \), you should recognize that:

• The expression involves multiplication, specifically multiplying the coefficients (numbers) and the powers of \( z \).
• Start with recognizing the consistent base, which here is \( z \).
• Understanding that you're dealing with similar bases can bring out the specific rule to simplify it, i.e., adding the exponents.

Recognizing the operation type is the first and vital step to guiding you towards applying the right methods to solve algebraic problems accurately.

Rules of Exponents

The rules of exponents are fundamental principles in algebra that guide us on how to handle and simplify expressions involving powers. One of these rules pertains to multiplying bases with exponents, known as the product of powers rule. This is crucial for simplifying expressions like the one in our example.
Here's a quick summary of key exponent rules relevant to multiplication:

• Product of Powers Rule: When multiplying two powers with the same base, add their exponents: \( a^m \times a^n = a^{m+n} \).
• Power of a Product Rule: To raise a power to another, multiply the exponents: \( (a^m)^n = a^{m\times n} \).
• Zero Exponent Rule: Any non-zero number raised to the power of zero is 1: \( a^0 = 1 \).

For example, applying the product of powers rule to \( z^3 \,\text{and} \, z^2 \), you add the exponents for a final result of \( z^5 \). This simplification process is all about applying these rules correctly to achieve cleaner and more manageable expressions.