Problem 28 - Intermediate Algebra

Question

Add or subtract. $$ 3 \sqrt{8 x^{2} y^{3}}-2 x \sqrt{32 y^{3}} $$

Step-by-Step Solution

Answer

The result is \(-2xy \sqrt{2y}\).

Step 1: Simplify the Radicals

Start by simplifying each radical expression individually. For the first term, \(3 \sqrt{8x^2y^3}\), notice that \(8 = 4 \times 2\) and \(y^3 = y^2 \times y\). Thus, \(\sqrt{8x^2y^3} = \sqrt{4\times 2x^2y^2\times y} = 2xy \sqrt{2y}\). Therefore, \(3 \sqrt{8x^2y^3} = 3 \cdot 2xy \sqrt{2y} = 6xy \sqrt{2y}\). For the second term, \(2x \sqrt{32y^3}\), notice that \(32 = 16 \times 2\). Thus, \(\sqrt{32y^3} = \sqrt{16 \times 2y^2 \times y} = 4y \sqrt{2y}\). Therefore, \(2x \sqrt{32y^3} = 2x \cdot 4y \sqrt{2y} = 8xy \sqrt{2y}\).

Step 2: Align Like Terms

Both terms are now \(xy \sqrt{2y}\) terms. It's essential to notice that they share the like term \(xy \sqrt{2y}\) which can be combined. This can be done through regular arithmetic on the coefficients.

Step 3: Perform Arithmetic on Coefficients

Subtract the coefficients of \(xy \sqrt{2y}\). You have \(6xy \sqrt{2y} - 8xy \sqrt{2y}\), which translates to \((6 - 8)xy \sqrt{2y} = -2xy \sqrt{2y}\).

Step 4: Combine and Conclude the Expression

The result from combining the like terms gives the final expression \(-2xy \sqrt{2y}\). This is the simplified result of the original expression.

Key Concepts

Simplifying Radicals

When dealing with radical expressions, simplifying the radicals is a crucial first step. Radical expressions often contain square roots or higher roots that can be broken down into simpler parts. This process involves factoring the number under the radical sign (also known as the radicand) into its prime factors.
For example, when you have a radicand like 8, realizing it as the product of perfect squares can help. For 8, we can write it as \(4 \times 2\). Here, 4 is a perfect square.

• If the radicand contains a variable like \(x^2\), it can also be broken down into roots.
• This principle is applied similarly to expressions like \(y^3\) splitting into \(y^2 \times y\).

Breaking down the radicand in this manner enables you to "take out" the square root of any perfect square part of the radicand, simplifying the radical expression as a whole.

Like Terms

Identifying and working with like terms is fundamental in algebraic simplification. Like terms are terms that contain the same variables raised to the same powers. In our example, both terms, after simplification, contained the expression \(xy \sqrt{2y}\).

• It’s important to note that the variables and the radical part must match exactly for the terms to be considered like terms.
• Numbers or coefficients in front of these expressions can differ but the algebraic and radical part must be the same.

After identifying like terms, you can combine them by focusing on adding or subtracting only their coefficients, making the simplification process straightforward.

Arithmetic on Coefficients

Performing arithmetic on coefficients is a simple yet crucial part of working with like terms. Once you have successfully identified like terms, you focus on their coefficients (the numerical part in front of the variable and radical expressions).

For the expression \(6xy \sqrt{2y} - 8xy \sqrt{2y}\), simply perform the arithmetic operation between the two numbers: 6 and 8.

• This operation results in \((6 - 8)\), which simplifies to -2.
• Then, attach this result back to the common algebraic term, which is \(xy \sqrt{2y}\).

The entire simplification boils down to shifting focus to the coefficients, performing basic arithmetic, and keeping the algebraic term consistent across the operation.

Square Roots in Algebra

Square roots are frequently encountered in algebra, especially when simplifying expressions or solving equations. A key aspect of working with square roots is recognizing and manipulating expressions containing them.

• Understanding that square roots can often be separated into parts involving perfect squares and non-perfect squares, as seen in \(\sqrt{8}\) becoming \(\sqrt{4 \times 2}\), is beneficial.
• Simplifying square roots involves extracting the square root of any perfect square factor. For instance, the \(\sqrt{4} = 2\), while the \(\sqrt{2}\) remains under the radical.

This simplification helps to make complex expressions more manageable, and is essential in more advanced algebraic operations.