Question

Combine radicals, if possible. $$ 4 x y \sqrt{90 x y}+2 \sqrt{40 x^{3} y^{3}}-3 x y \sqrt{50 x y} $$

Short answer

Question: Simplify the expression \(4xy\sqrt{90xy} + \sqrt{40x^3y^3} - 3xy\sqrt{50xy}\) and combine like terms if possible. Answer: \(14x^2y^2\sqrt{10} - 15x^2y^2\sqrt{2}\)

Step-by-step solution

Simplify the radical terms

First, factor out the perfect squares in each radicand. Remember that \(\sqrt{ab} = \sqrt{a}\sqrt{b}\) for any positive numbers a and b. For the first term: \(\sqrt{90xy} = \sqrt{9 \cdot 10 \cdot x \cdot y} = \sqrt{9x} \sqrt{10y} = 3\sqrt{x} \cdot \sqrt{10y} = 3\sqrt{x} \sqrt{y} \sqrt{10}\). So, \(4xy\sqrt{90xy} = 4xy \cdot 3\sqrt{x}\sqrt{y}\sqrt{10} = 12x^2y^2\sqrt{10}\). For the second term: \(\sqrt{40x^3y^3} = \sqrt{4 \cdot 10 \cdot x^3 \cdot y^3} = \sqrt{4x^2} \sqrt{10y^2} \sqrt{x} \sqrt{y} = 2xy\sqrt{10}\sqrt{x}\sqrt{y} = 2x^2y^2\sqrt{10}\) For the third term: \(\sqrt{50xy} = \sqrt{25 \cdot 2 \cdot x \cdot y} = \sqrt{25x} \sqrt{2y} = 5\sqrt{x} \cdot \sqrt{2y} = 5\sqrt{x}\sqrt{y}\sqrt{2}\), So \(-3xy\sqrt{50xy} = -3xy \cdot 5\sqrt{x}\sqrt{y}\sqrt{2} = -15x^2y^2\sqrt{2}\). Now, rewrite the expression using the simplified terms: $$ 12x^2y^2\sqrt{10} + 2x^2y^2\sqrt{10} - 15x^2y^2\sqrt{2} $$

Combine like terms

Now, we can combine the terms with the same radicals: $$ (12x^2y^2 + 2x^2y^2)\sqrt{10} + (-15x^2y^2)\sqrt{2} $$ Simplify the coefficients: $$ 14x^2y^2\sqrt{10} - 15x^2y^2\sqrt{2} $$ This is the final combined expression.

Key concepts

Combining Like Terms

When working with expressions that involve radicals, combining like terms is an essential step to simplify the expression. But what exactly does "combining like terms" mean in the context of radical expressions? Simply put, like terms are terms that have the same variables raised to the same powers and, in the case of radicals, have identical radicands (the number or expression inside the radical sign).

• For example, in the expression \( 12x^2y^2\sqrt{10} + 2x^2y^2\sqrt{10} \), the terms are considered like terms because they both include the \( \sqrt{10} \) and the variable part \( x^2y^2 \).
• However, in the term \( -15x^2y^2\sqrt{2} \), the radicand \( \sqrt{2} \) is different, which means it isn't like the others.

To combine like terms, you add or subtract their coefficients while keeping the rest of the term identical. So, for \( 12x^2y^2\sqrt{10} + 2x^2y^2\sqrt{10} \), you sum \( 12 \) and \( 2 \) to get \( 14 \), resulting in \( 14x^2y^2\sqrt{10} \). There is no possibility to combine it with \( -15x^2y^2\sqrt{2} \) due to the differing radicals.

Simplifying Radicals

Simplifying radicals is a crucial step when working with radical expressions, as it makes subsequent algebraic manipulation and combining terms more straightforward. Here's how you can simplify radicals:

• First, identify the factors within the radicand that can be extracted as perfect squares. For instance, in \( \sqrt{90xy} \), 90 can be factored as \( 9 \times 10 \), and \( 9 \) is a perfect square.
• The square root of 9 is 3, so \( \sqrt{90xy} = \sqrt{9 \times 10 \times x \times y} = 3\sqrt{10xy} \).
• This process is repeated for all terms to express them in their simplest radical form, as seen with \( \sqrt{40x^3y^3} \) and \( \sqrt{50xy} \) in the solution.

By breaking down the radical into its simplest form, you ensure that each term is as simplified as possible, preparing it for combining with like terms in the expression.

Algebraic Manipulation

Algebraic manipulation involves reorganizing and modifying expressions using algebraic rules to reach a simpler or more useful form. In dealing with radicals, especially, manipulation involves several stages:

• Initially, simplify each radical to its simplest form. This was done by extracting and separating perfect square factors.
• Next, identify and group like terms using their coefficients and radicals. Be sure to pay close attention to the variables and radicals involved.
• Then, perform addition or subtraction on the coefficients of these terms to come up with combined results for each group.

Once you've successfully managed these operations, you achieve a streamlined expression. For our example, the expression was manipulated to a cleaner form by grouping and combining like terms resulting in \( 14x^2y^2\sqrt{10} \) while accounting for separate terms \( -15x^2y^2\sqrt{2} \). This skill is invaluable for simplifying complex algebraic expressions and solving equations.