Problem 4 - Elementary Algebra

Question

Make use of either or both the power rule for products and the power rule for powers to simplify each expression. $$ (3 b x y)^{2} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \((3 b x y)^{2}\). Answer: \(9b^2x^2y^2\)

Step 1: Understand Properties Used

To simplify the given expression, we will mainly use the power rule for products. This rule states that the exponent "n" applies to each term separately. For instance, \((ab)^n = a^n b^n\). Now let's apply this rule to the given expression.

Step 2: Apply the Power Rule for Products

Using the power rule for products, we distribute the exponent of 2 to each term within the parentheses. So, the given expression simplifies as follows: \((3 b x y)^{2} = 3^2 b^2 x^2 y^2\)

Step 3: Simplify the Numerical Value

Now, we will simplify the numerical value (\(3^2\)). To do this, multiply 3 by itself: \(3^2 = 3*3 = 9\)

Step 4: Write the final expression

By putting everything together, we get the final simplified expression: \((3 b x y)^2 = 9b^2x^2y^2\) So, the simplified expression of \((3 b x y)^{2}\) is \(9b^2x^2y^2\).

Key Concepts

Understanding the Power Rule

The power rule is a fundamental concept in mathematics, particularly useful in algebra and calculus. When dealing with expressions like \((3bxy)^2\), the power rule helps us distribute the exponent across each element within the parenthesis.

The power rule for products asserts that if you have an expression of the form \((ab)^n\), you can apply the exponent \(n\) to each individual element, resulting in \(a^n\) and \(b^n\). This rule is extremely helpful for breaking down complex expressions into more manageable pieces.

• For the term \(3\), applying the exponent gives \(3^2\).
• For the term \(b\), it becomes \(b^2\).
• Similarly, \(x\) and \(y\) become \(x^2\) and \(y^2\) respectively.

Once the power rule is applied, the expression becomes a series of terms raised to the given power, making the process of simplification much smoother and organized.

The Art of Simplification

Simplification is the process of rewriting an expression in its simplest form; a skill which is vital for solving algebraic equations efficiently.

In the exercise, after applying the power rule, we have the expression \(3^2 b^2 x^2 y^2\). Simplification mainly involves calculating powers and combining like terms. Calculating \(3^2\) is the first step, which simplifies to 9 as you multiply 3 by itself (\(3 \times 3\)).

• The numerical calculation simplifies \(3^2\) to 9.
• No combination of like terms is needed in this instance, since all terms are distinct.

The final simplified expression is \(9b^2x^2y^2\), which represents a much more straightforward version of the original problem.

Exploring Exponents

Exponents are a shorthand for denoting repeated multiplication of a number by itself; they play a crucial role in expressing large numbers in a more concise form.

Understanding exponents is fundamental to manipulating algebraic expressions. The notation \(a^n\) means that the base \(a\) is multiplied by itself \(n\) times. For instance, \(3^2\) denotes multiplying 3 by itself resulting in 9.

• Exponents tell us how many times to multiply a number.
• They allow for easier manipulation of polynomial terms by providing a common understanding of multiplication operations needed.

Recognizing how exponents work helps students simplify expressions efficiently, especially when combined with power rules and other algebraic techniques.