Problem 26 - Algebra A Combined Function

Question

Use the product rule to simplify each expression. $$ \left(-2 z^{3}\right)\left(-2 z^{2}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \( 4z^5 \).

Step 1: Identify the Product Rule

The product rule states that when multiplying two terms of the same base, you add the exponents. The formula is given by: if you have terms like \( a^m \) and \( a^n \), then \( a^m \cdot a^n = a^{m+n} \). In this case, our variable base is \( z \).

Step 2: Simplify the Coefficients

There are two coefficients in the expression: \(-2\) and \(-2\). Multiply these coefficients together. \(-2 \times -2 = 4\).

Step 3: Simplify the Exponential Part

Apply the product rule to the exponents of \( z \). Since you have \( z^3 \) from the first term and \( z^2 \) from the second term, you add the exponents: \( 3 + 2 = 5 \). Thus, \( z^{3+2} = z^5 \).

Step 4: Combine Results

Combine the results from the simplified coefficients and the exponentiated variable. The expression \( 4z^5 \) is obtained by combining 4 from the coefficients and \( z^5 \) from the exponential part.

Key Concepts

Exponents

Exponents are a way to express repeated multiplication of a number by itself. In algebra, we use exponents to easily manage and compute large numbers. When we have a base raised to a power, as in \( z^3 \), it means we multiply \( z \) by itself three times: \( z \times z \times z \). The product rule in exponents tells us how to handle multiplication when dealing with the same base.

• With the same base, you add the exponents: \( a^m \cdot a^n = a^{m+n} \).
• This rule is incredibly useful for simplifying expressions quickly.

In our case, when given the problem \( (-2z^3) \times (-2z^2) \), the exponents of \( z \) are 3 and 2. Applying the rule, you add 3 and 2 to get 5. Thus, the exponent part becomes \( z^5 \). This makes calculations simpler and neat.

Coefficient Simplification

Coefficients in algebra are the numerical parts of the terms. In the expression \( -2z^3 \times -2z^2 \), the coefficients are both \(-2\).

• To simplify the expression, you need to multiply these coefficients.
• Multiplying \(-2 \times -2\) gives us 4. Remember, multiplying two negative numbers results in a positive.

Coefficient simplification helps you separate the numerical multiplication from the variable part, making it easier to handle each part of an algebraic expression separately.
After simplifying the coefficients and the exponents, you can then combine both results to give the final expression.
This way, you understand each step and how every component contributes to the final simplified answer.

Multiplying Algebraic Expressions

Multiplying algebraic expressions involves both dealing with coefficients and managing variable exponents. The task is often simplified using specific rules to break down the process into manageable parts. Consider the original expression \( (-2z^3) \times (-2z^2) \).

• First, focus on coefficients: \(-2\) and \(-2\). Multiply to get 4.
• Next, shift attention to the variables with exponents. Use the product rule to handle the \(z\) terms: \(z^3\) and \(z^2\) add to \(z^{3+2} = z^5\).

Together, the simplified coefficient 4 and the combined exponent \(z^5\) form the final expression \(4z^5\).
By systematically addressing each part of the multiplication, the process becomes straightforward, resulting in a simplified expression clear of any unnecessary complexity.