Problem 61 - Algebra for College Students

Question

Express each of the following in simplest radical form. All variables represent positive real numbers. \(\sqrt{8 x+12 y} \quad[\) Hint: \(\sqrt{8 x+12 y}=\sqrt{4(2 x+3 y)}]\)

Step-by-Step Solution

Answer

The simplest radical form is \(2\sqrt{2x+3y}\).

Step 1: Simplify the Factor Inside the Square Root

The expression given inside the square root can be expressed as \(\sqrt{8x + 12y}\). The hint suggests factoring out a 4: \(\sqrt{4(2x+3y)}\). This helps in simplifying the expression further, as it can help in taking advantage of the property of square roots that allows us to separate the square root of products.

Step 2: Apply the Property of Square Roots

The square root of a product, \(\sqrt{ab}\), can be expressed as a product of square roots: \(\sqrt{a}\cdot\sqrt{b}\). Using this property, \(\sqrt{4(2x+3y)}\) can be split into \(\sqrt{4}\cdot\sqrt{2x+3y}\).

Step 3: Calculate the Square Root of the Factor

Now, calculate the square root of 4. The square root of 4 is 2. Thus, the expression \(\sqrt{4}\cdot\sqrt{2x+3y}\) simplifies further to \(2\sqrt{2x+3y}\). This is because taking the square root of a perfect square (like 4) results in its base number (in this case, 2).

Key Concepts

Radical Expressions

Radical expressions appear frequently in mathematics and involve finding the root of a number. Typically, these are square roots, but could be any root such as cube roots. The word "radical" is derived from "radix", which means "root". Simplifying radicals involves rewriting them in their simplest form so they can be understood and used more easily in equations or expressions, like converting \(\sqrt{8x + 12y}\) to its simplest form. This process can make computations easier and clearer to follow.

• A radical expression generally looks like this: \(\sqrt[n]{a}\), where \(n\) is the degree of the root (2 for square roots).
• Simplifying radicals often involves expressing the number inside the root in terms of a product of its factors, leveraging certain algebraic properties to simplify the calculations.

This forms the backbone of tackling more complex algebraic problems and helps in maintaining consistency and accuracy when performing operations with these numbers.

Factoring

Factoring is an important mathematical strategy that refers to breaking down an expression into products of other expressions. This is useful when simplifying radical expressions, as seen in the expression \(\sqrt{8x + 12y}\). By factoring, you can often pull out perfect squares which can directly influence how the radical might be simplified.

• In the given example, the hint suggests factoring out a 4 from \(8x + 12y\), simplifying it to \(\sqrt{4(2x+3y)}\).
• This process aids in identifying components in the expression that can be easily manipulated using algebraic properties.

Understanding how to efficiently factor expressions is crucial in simplifying radicals and is a common skill used in algebra to simplify expressions and solve equations more efficiently.

Properties of Square Roots

Square roots have specific properties that allow mathematicians to simplify expressions involving them. A key principle is that the square root of a product can be expressed as the product of square roots: \(\sqrt{a imes b} = \sqrt{a} \cdot \sqrt{b}\). This property is particularly useful when dealing with expressions like \(\sqrt{4(2x+3y)}\).

• Using this property, \(\sqrt{4(2x+3y)}\) can be split into two separate square roots: \(\sqrt{4} \cdot \sqrt{2x+3y}\).
• Square roots of perfect squares result in whole numbers, making calculations easier. For instance, \(\sqrt{4} = 2\).

Leveraging these properties will streamline the process of solving and simplifying radical expressions, forming a fundamental part of solving many algebraic problems.