Problem 26 - Prealgebra

Question

Simplify each expression by taking as much out from under the radical as possible. You may assume that all variables represent positive numbers $$\sqrt{72 x^{4} y^{2}}$$

Step-by-Step Solution

Answer

The simplified expression is \(6x^2y\sqrt{2}\).

Step 1: Break Down the Expression Under the Radical

Start by factoring the components inside the radical. Rewrite the expression under the square root as a product of perfect squares. The expression \(72x^4y^2\) can be broken down to \(36 \times 2 \times x^4 \times y^2\). Notice that \(36\) is a perfect square, as are \(x^4\) and \(y^2\).

Step 2: Simplify the Perfect Squares

Next, take the square root of each perfect square component. The square root of \(36\) is \(6\), the square root of \(x^4\) is \(x^2\), and the square root of \(y^2\) is \(y\).

Step 3: Simplifying the Remaining Radical

You are left with \(\sqrt{2}\) under the radical since 2 is not a perfect square. The expression at this point is \(6x^2y\sqrt{2}\).

Step 4: Combine the Simplified Parts

Combine all the parts to express the entire radical expression in simplified form. The final simplified expression is \(6x^2y\sqrt{2}\).

Key Concepts

Factoring

Factoring is like breaking a number or a mathematical expression into parts, called factors, that can be multiplied to produce the original number or expression. In our exercise, the expression under the radical is \(72x^4y^2\). First, we need to factor this composite number to identify any perfect squares.

• Consider the number 72, which factors into \(36 \times 2\). Here, 36 is a perfect square \((6^2)\).
• The exponent \(x^4\) can be rewritten as \((x^2)^2\), indicating that it is also a perfect square.
• Similarly, \(y^2\) is already a perfect square \((y)^2\).

By factoring in such a manner, we set the stage to simplify the expression going forward.

Perfect Squares

A perfect square is a number or a variable that can be expressed as the product of an integer with itself. Perfect squares play a crucial role in simplifying radical expressions as they allow us to "escape" the square root.

• In our example, 36 is a perfect square since \(6 \times 6 = 36\). Thus, the square root of 36 is simply 6.
• The variable \(x^4\) is written as \((x^2)^2\), making \(x^2\) its perfect square root.
• Similarly, \(y^2\) is already a perfect square, simplifying directly to \(y\).

By identifying these perfect squares, we simplify parts of an expression to make it more manageable, reducing complexity in algebraic manipulations.

Radicals

Radicals, often known as roots, represent the opposite operation of raising a number to a power. The most common radical is the square root, denoted by the symbol \(\sqrt{}\). Simplifying expressions involving radicals involves multiple steps:

• Identify any perfect squares under the radical since these can be swiftly simplified.
• The component that cannot be expressed as a perfect square, like \(\sqrt{2}\) in our example, remains under the radical symbol.
• Simplified forms like \(6x^2y\sqrt{2}\) emerge from breaking down and rebuilding the radical into simpler parts.

Understanding radicals lets us simplify complex expressions into neater, more understandable forms.

Algebra Basics

The foundations of algebra help us tackle expressions like \(\sqrt{72x^4y^2}\) with ease. Familiarity with core operations such as factoring, recognizing perfect squares, and working with radicals are all elements involved.

• Recognition of components that can be simplified, such as integers and powers, is essential.
• Simplification involves the gradual breakdown of expressions into basic parts, then assembling them into simpler solutions.
• Mastering these basics improves problem-solving capabilities and enhances mathematical fluency across a wide range of concepts.

Building a strong foundation in algebra allows for seamless navigation through more complex equations and scenarios, ultimately leading to greater mathematical success.