Problem 43 - Elementary Algebra

Question

Simplify. $$ \sqrt{144 x^{2} y^{2}} $$

Step-by-Step Solution

Answer

12xy

Step 1 - Identify the Square Root

Recognize that the root function should be applied to each part of the expression inside the square root, which is \( 144 x^2 y^2 \).

Step 2 - Split the Terms

Split the terms inside the square root: \( \ \sqrt{144 x^2 y^2} = \ \sqrt{144} \ \sqrt{x^2} \ \sqrt{y^2} \ \)

Step 3 - Simplify Each Term

Simplify each term separately: \ \( \ \sqrt{144} = 12, \ \sqrt{x^2} = x, \ \sqrt{y^2} = y \ \)

Step 4 - Combine the Simplified Terms

Combine the simplified terms: \ \( 12 \ x \ y \ \)

Key Concepts

Simplifying Square Roots

Square root simplification is a fundamental skill in algebra. It involves breaking down each term inside a square root to simplify the overall expression. For example, in the exercise given, we have the expression \( \sqrt{144 x^2 y^2} \).

• Identify the parts inside the square root.
• Apply the square root function to each part individually.
• Simplify each of these parts separately.

In our example, \( \sqrt{144 x^2 y^2} \), we split this into \( \sqrt{144} \sqrt{x^2} \sqrt{y^2} \). Next, simplify each term: \( \sqrt{144} = 12 \), \( \sqrt{x^2} = x \), and \( \sqrt{y^2} = y \). Finally, combine the simplified results to get \( 12xy \). Simplifying square roots makes complex algebraic expressions much more manageable.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. In algebra, we often need to manipulate these expressions for different purposes, such as solving equations or simplifying terms.
Consider the expression \( 144x^2y^2 \). Here:

• 144 is a constant coefficient.
• \(x^2\) and \(y^2\) are variables raised to powers.

To simplify expressions that include square roots, factorize the terms as much as possible. For instance, recognize that \( 144 = 12^2 \), \( x^2 \) and \( y^2 \) are already perfect squares. Taking the square root of each part simplifies the expression. This principle is widely used to break down and solve more complex algebraic challenges.
Simplifying algebraic expressions involves recognizing patterns and applying the correct operations sequentially.

Basic Algebra

Basic algebra involves fundamental concepts such as handling variables, constants, and arithmetic operations. Mastering these basics is crucial for tackling more complicated problems.
In our exercise, we applied several basic algebra principles:

• The square root property: \( \sqrt{a^2} = a \).
• Combining like terms correctly after simplification.

By breaking down the expression \( \sqrt{144x^2y^2} \), we applied basic rules step-by-step: recognize the square terms, simplify each term, and then recombine them correctly.
These principles help build a solid foundation in algebra, allowing more advanced mathematical concepts to be more easily understood and applied.