Problem 26 - Algebra for College Students

Question

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. \(4 \sqrt{x}(2 \sqrt{x y}+2 \sqrt{x})\)

Step-by-Step Solution

Answer

The simplest radical form is \(8x \sqrt{y} + 8x\).

Step 1: Distribute the First Term

Use the distributive property to multiply the first term outside the parentheses by each term inside. This means you'll multiply \(4 \sqrt{x}\) by \(2 \sqrt{xy}\) and \(4 \sqrt{x}\) by \(2 \sqrt{x}\).

Step 2: Multiply the First Pair

Multiply \(4 \sqrt{x}\) by \(2 \sqrt{xy}\). This gives: \(4 \times 2 \times \sqrt{x} \times \sqrt{xy} = 8 \sqrt{x \cdot xy} \). Simplify to get \(8 \sqrt{x^2y} \).

Step 3: Simplify the First Product

Realize that \( \sqrt{x^2y} = x \sqrt{y} \). So, \(8 \sqrt{x^2y} = 8x \sqrt{y} \).

Step 4: Multiply the Second Pair

Multiply \(4 \sqrt{x}\) by \(2 \sqrt{x}\). This results in \(8 \sqrt{x \cdot x} = 8 \sqrt{x^2}\).

Step 5: Simplify the Second Product

Since \( \sqrt{x^2} = x \), then \(8 \sqrt{x^2} = 8x \).

Step 6: Combine Like Terms

Combine the results from Step 3 and Step 5 to get the final expression. Add \(8x \sqrt{y} + 8x\) to simplify the expression to its simplest form. So, the final expression is \(8x \sqrt{y} + 8x\).

Key Concepts

distributive property

The distributive property is a useful tool in algebra that helps us expand expressions involving parentheses. It states that for any numbers or expressions, \( a(b + c) = ab + ac \). This property allows you to multiply the term outside the parentheses by each term inside.
In our exercise, we apply the distributive property by taking the term outside, which is \( 4 \sqrt{x} \), and multiplying it with each of the terms inside the parentheses, \( 2 \sqrt{xy} \) and \( 2 \sqrt{x} \).
By doing so, we break down the original expression into simpler parts. This step is often the first in solving more complex problems, making it an essential concept to master in algebra.

multiplication of radicals

Multiplying radicals involves a few straightforward steps. First, if the radicals have the same index, you multiply the integers or coefficients in front of the radicals. Then, you multiply the radicands—the numbers inside the square root.
For example, we multiply \( 4 \sqrt{x} \) by \( 2 \sqrt{xy} \) to get \( 8 \sqrt{x \cdot xy} \). It's important to remember to combine like terms, such as the numbers outside the radicals, separately from the radicands.
Properly applying these steps allows us to simplify complex expressions involving radicals. This makes working with them much easier and less intimidating.

simplifying square roots

Simplifying square roots involves expressing the square root in its simplest form. A key idea is finding perfect squares within the radicand. For instance, \( \sqrt{x^2 y} \) can be broken down to \( x \sqrt{y} \), since \( x^2 \) is a perfect square.
When simplifying, identify any perfect square factors. Remove these factors from the square root, leaving other factors inside. For example, \( \sqrt{x^2} = x \), since the square root of a perfect square yields the base number itself.
Understanding how to simplify square roots is important because it helps reduce expressions to their most basic form, making them easier to work with or combine in further calculations.

algebraic expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operators. In simple terms, it is a combination of terms formed by numbers and letters representing values.
In expressions like \( 8x \sqrt{y} + 8x \), each part—\( 8x \sqrt{y} \) and \( 8x \)—is a term. Terms are combined by addition or subtraction in an expression.
Understanding algebraic expressions is crucial for solving equations and simplifying mathematical problems. They form the foundation of algebra, which is used in various fields, from calculating financial formulas to solving engineering problems.