Problem 17 - Elementary Algebra

Question

For the following problems, expand each product (do not compute the actual value). \(x^{4} \cdot y^{4}\)

Step-by-Step Solution

Answer

Answer: The expanded form of the expression is \(x^{4}y^{4}\).

Step 1: The given expression is \(x^{4} \cdot y^{4}\). To expand the product, we can use the laws of exponents, which states that when multiplying the same base raised to different powers, we simply add the exponents respectively. However, in this case, the bases are different (x and y). Therefore, no further simplification can be made. #Step 2: Write the expanded form

Since there is no further simplification to be made, the expanded form of the given expression is simply \(x^{4} \cdot y^{4}\). The final expanded form of the expression is \(x^{4}y^{4}\).

Step 2: Identify the operation required

Determine what computation or simplification is needed.

Step 3: Perform the calculation

Execute the computation step by step.

Step 4: Simplify and state the result

Simplify and write the final answer.

Step 5: Conclude with the answer

Answer: The expanded form of the expression is \(x^{4}y^{4}\).

Key Concepts

Laws of Exponents

The laws of exponents are essential rules that help us work with powers in mathematical expressions. They simplify complex calculations and provide a clear path forward when solving exponent problems. Here’s a brief look at key exponent laws:

• Product of Powers: When you multiply two powers with the same base, you add the exponents, like this: \(a^m \cdot a^n = a^{m+n}\).
• Quotient of Powers: Dividing? Subtract the exponents: \(a^m / a^n = a^{m-n}\).
• Power of a Power: Raise a number that has an exponent, and you multiply them: \((a^m)^n = a^{m \cdot n}\).

These rules make it easier to handle expressions, but it’s crucial to note the base must be the same when applying these rules. This distinction is critical, especially since our exercise didn't allow us to simplify further due to differing bases.

Multiplication of Powers

Understanding the multiplication of powers becomes straightforward once we grasp the laws of exponents. Multiplying powers involves working with the exponents efficiently. Here’s how:

• When the base is the same, simply add the exponents: \(a^m \cdot a^n = a^{m+n}\).
• If the bases are different, as seen in our exercise with \(x^4 \cdot y^4\), the rule isn't applied, and the expression remains unchanged.

The key is recognizing when these laws are applicable. The exercise demonstrates a crucial teaching moment about different bases. When bases differ, each part stands alone, maintaining its initial power.

Algebraic Expressions

Algebraic expressions encompass numbers, variables, and operations, including powers or exponents. They are the building blocks of algebra and often involve several different components:

• Terms: Parts of an expression separated by addition or subtraction signs, like \(3x^2\) or \(-5y\).
• Variables: Symbols representing numbers we don't yet know, shown often as letters like \(x\), \(y\).
• Coefficients: Numbers in front of the variables, signaling how many times the term is multiplied. For example, in \(4x\), the number 4 is the coefficient.

Recognizing how terms and components interact is crucial for understanding and manipulating algebraic expressions. For example, in the problem \(x^4 \cdot y^4\), the expression is broken into parts, and each power remains unaffected by the other's presence. Understanding these components helps when expanding, simplifying, or rearranging expressions.