Problem 38 - Algebra 1

Question

Write your answer as a power or as a product of powers. $$ (-3 a)^{5} \cdot(4 a)^{2} $$

Step-by-Step Solution

Answer

\(-3888a^7\)

Step 1: Apply Exponent to Each Factor Inside Parentheses

The first step is to apply the exponents to each factor inside the parentheses. We have \( (-3)^5 \cdot a^5 \cdot (4)^2 \cdot a^2 \). This simplifies into \( -243 a^5 \cdot 16 a^2 \).

Step 2: Combine Like Terms

As per the multiplication rule of exponents, when we multiply like bases, we add the exponents. Therefore, \(a^5\) multiplied by \(a^2\) is \(a^{5+2}\) or \(a^7\). Also, the numerical factors will be multiplied together to get -3888.

Step 3: Finalize the Expression

After combining like terms, the expression simplifies to \(-3888a^7\).

Key Concepts

Multiplication of Like Bases

Understanding how to multiply terms with the same base is crucial in simplifying expressions involving exponents. When we have like bases – meaning the same variable or number – raised to different powers, we add the exponents together. This process is governed by the law of exponents:

• For example, if you have \( a^m \times a^n \), it simplifies to \( a^{m+n} \).

In the original exercise, we had \( a^5 \) and \( a^2 \). Applying the rule of multiplication for like bases, \( a^5 \cdot a^2 \) becomes \( a^{5+2} \) or \( a^7 \). This rule greatly simplifies complex expressions. Make sure to keep track of which terms share bases to avoid errors when adding exponents.

Exponentiation

Exponentiation is the mathematical operation involving numbers raised to a power. It’s a shorthand for repeated multiplication of the same factor. For instance:

• \( x^3 \) means \( x \times x \times x \).
• \( y^2 \) means \( y \times y \).

In the original exercise, each term inside the parentheses had to be exponentiated. This meant raising both numbers and variables to the respective powers, i.e., \( (-3)^5 \) and \( 4^2 \). An important note is the handling of negative bases: \( (-3)^5 \) results in a negative product since the exponent is odd (an even exponent would result in a positive product). Remembering these nuances protects against sign mistakes in your calculations.

Simplifying Expressions

Simplifying expressions isn't just about solving them; it's about making them as straightforward as possible. This involves combining like terms and using the properties of exponents.

• First, simplify each component of the expression by applying the powers to numbers and variables.
• Next, combine the numerical results and variables separately, keeping track of the signs and applying multiplication as necessary.

In our example: after exponentiating, we needed to multiply \(-243\) and \(16\) to get \(-3888\). Similarly, combining \( a^5 \) and \( a^2 \) becomes \( a^7 \). Putting these together, the final simplified expression is \(-3888a^7\). Simplifying expressions involves both following steps procedurally and occasionally re-checking each step for accuracy.