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Chapter 7: Problem 67

Multiply out each of the following. As you work out the problems, identify those exercises that are either a perfect square or the difference of two squares. $$(3 x y)^{2}$$

Short Answer

Expert verified
The simplified expression is \( 9x^2y^2 \). It is a perfect square.

Step by step solution

01

Identify the expression

The expression given is \( (3xy)^2 \). It is a single term raised to the power of 2.
02

Apply the exponentiation rule

Use the rule \( (a \times b)^n = a^n \times b^n \), where \(a = 3xy\) and \(n = 2\). Thus, \( (3xy)^2 = 3^2 \times x^2 \times y^2 \).
03

Calculate the powers

Calculate each term separately: \(3^2 = 9\), \(x^2\) remains \(x^2\), and \(y^2\) remains \(y^2\). So, \(9 \times x^2 \times y^2 = 9x^2y^2\).
04

Simplify the expression

Combine the terms to get the simplified expression: \(9x^2y^2\).
05

Identify the type of expression

Observe that \(9x^2y^2\) is a perfect square because it can be written in the form \(a^2\), where \(a = 3xy\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perfect Square
In algebra, a perfect square is a product of a number multiplied by itself. This concept helps simplify complex algebraic expressions. In our exercise, the expression \((3xy)^2\) represents a perfect square.
Exponentiation Rule
The exponentiation rule is key in algebra. It states that \( (a \times b)^n = a^n \times b^n \). Using this rule makes working with powers easier. When we applied it to \((3xy)^2\), we transformed it into \(3^2 \times x^2 \times y^2 = 9x^2y^2 \). This rule breaks down complex terms, helping simplify the process.
Multiplication
Multiplication of algebraic expressions involves dealing with coefficients and variables. Here, \(3^2 = 9\), and the variables remain as \(x^2\) and \(y^2\). Combining them, we get \(9x^2y^2\). This process shows how multiplication affects both numbers and variables in algebraic contexts.

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