Problem 67 - Elementary and Intermediate Algebra

Question

Use the power of a product rule for exponents to simplify each expression. $$ (6 a)^{2} $$

Step-by-Step Solution

Answer

The expression \((6a)^2\) simplifies to \(36a^2\).

Step 1: Identify the Power of a Product Rule

The power of a product rule states that for any numbers or expressions \(a\) and \(b\), and any integer \(n\), \((ab)^n = a^n b^n\). We will apply this rule to the expression \((6a)^2\).

Step 2: Apply the Power to Each Factor

According to the power of a product rule, we distribute the square to both the numerical coefficient and the variable: \((6a)^2 = 6^2 \cdot a^2\).

Step 3: Calculate the Powers of Each Factor

Now, we calculate each power: \(6^2 = 36\) and \(a^2 = a^2\). Therefore, the expression becomes: \(36 \cdot a^2\).

Step 4: Final Step: Write the Simplified Expression

Combine the calculated parts to write the final simplified expression. Thus, \((6a)^2\) simplifies to \(36a^2\).

Key Concepts

Exponents

Exponents are a way of expressing repeated multiplication of a number by itself. Essentially, when you see a number raised to a power, it tells you how many times to multiply the number by itself. For instance, in the expression \(6^2\), the base is 6, and the exponent is 2, meaning you multiply 6 by itself: \(6 \times 6 = 36\). This concept doesn't change when we apply it to variables like in \(a^2\), which simply means \(a \times a\).
Exponents follow specific arithmetic rules that help simplify expressions. Keep these key points in mind:

• The base number or expression is the number you're multiplying.
• The exponent tells you how many times to multiply the base by itself.
• Any number or variable to the power of 1 is just itself, \(a^1 = a\).

Understanding these rules is crucial when dealing with expressions that involve both numbers and variables.

Simplifying Expressions

Simplifying expressions makes them more manageable. It's like cleaning up a messy room, where grouping similar items together in a neat order helps the room make more sense. Similarly, simplifying allows you to organize and reduce expressions into their simplest form.
Let's take the example of simplifying \((6a)^2\). We use the power of a product rule, which allows us to apply the exponent to each factor of the product separately. Here, the product \((6a)\) consists of a number, 6, and a variable, \(a\).
By applying the power to each component:

• We calculate \(6^2\), which simplifies to 36.
• We then calculate \(a^2\), leaving it as \(a \times a\).

Finally, combining the results gives us the simplified expression \(36a^2\). This method of simplifying expressions keeps calculations straightforward and makes complex algebra much easier.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that together form a mathematical phrase. They are like the language of mathematics, allowing us to describe and manipulate numerical relationships.
An expression can consist of:

• A single number (constant), like 6.
• A single variable, like \(a\).
• Or a combination of numbers, variables, and arithmetic operations, such as \(6a\).

When simplifying an algebraic expression, one key strategy is to apply rules like the power of a product. This rule is used when you encounter terms like \((xy)^n\), which becomes \(x^n \times y^n\). Applying these rules to algebraic terms ensures that each part of the expression is properly handled, making the expression easier to work with in equations and real-world applications.