Problem 65 - Algebra for College Students

Question

For Problems \(65-74\), use the distributive property to help simplify each of the following. All variables represent positive real numbers. \(-3 \sqrt{4 x}+5 \sqrt{9 x}+6 \sqrt{16 x}\)

Step-by-Step Solution

Answer

The simplified expression is \(33\sqrt{x}\).

Step 1: Understanding the Problem

The expression given is \(-3 \sqrt{4x} + 5 \sqrt{9x} + 6 \sqrt{16x}\). Each term involves a square root of a product involving the expression \(k\sqrt{c x}\), where \(k\) is a coefficient and \(c\) is a perfect square.

Step 2: Simplify Each Square Root Component

Start by simplifying the square root in each term individually. For \( \sqrt{4x} \), this becomes \( \sqrt{4}\sqrt{x} = 2\sqrt{x} \). Similarly, \( \sqrt{9x} = 3\sqrt{x} \) and \( \sqrt{16x} = 4\sqrt{x} \).

Step 3: Apply the Distributive Property

Using the distributive property, substitute back into the expression: \(-3(2\sqrt{x}) + 5(3\sqrt{x}) + 6(4\sqrt{x})\). This results in \(-6\sqrt{x} + 15\sqrt{x} + 24\sqrt{x}\).

Step 4: Combine Like Terms

Combine the coefficients of \( \sqrt{x} \). First, find \(-6\sqrt{x} + 15\sqrt{x}\), which equals \(9\sqrt{x}\). Then add this to \(24\sqrt{x}\), resulting in \(33\sqrt{x}\).

Step 5: Conclude the Simplification

The simplified expression is \(33\sqrt{x}\). All the square root terms combined using their coefficients give the final simplified result.

Key Concepts

Simplifying Radical Expressions

Simplifying radical expressions involves breaking down and simplifying square roots or other root expressions. To simplify a radical expression like the one in the problem, the first step is ensuring that each square root part involves a perfect square factor that can be simplified easily.
For instance, in the expression \(-3 \sqrt{4x} + 5 \sqrt{9x} + 6 \sqrt{16x}\):

• The radical part \( \sqrt{4x} \) simplifies to \( \sqrt{4}\sqrt{x} \), which is equal to \( 2\sqrt{x} \) since \( \sqrt{4} = 2 \).
• Similarly, \( \sqrt{9x} \) becomes \( 3\sqrt{x} \), as \( \sqrt{9} = 3 \).
• And \( \sqrt{16x} \) is simplified to \( 4\sqrt{x} \), given \( \sqrt{16} = 4 \).

By identifying and extracting the perfect squares, we can write each term in a simpler form, which makes it easier to work with in subsequent steps of the problem.

Combining Like Terms

Combining like terms is a crucial part of simplifying expressions. Once you have simplified the individual radicals, each term in the expression becomes more manageable.
In our expression: \(-6\sqrt{x} + 15\sqrt{x} + 24\sqrt{x}\), all terms include \( \sqrt{x} \) as a common factor, making them like terms.

• First, add the coefficients of the like terms together: \(-6 + 15 + 24\).
• This summation results in a total coefficient of \(33\).
• Since they all share \( \sqrt{x} \) as a factor, you can combine them as \( 33\sqrt{x} \).

This technique of combining like terms reduces a complex expression into a simpler one, making it much easier to handle and understand.

Square Roots

Square roots are fundamental in many areas of mathematics, especially in simplifying radical expressions. The square root of a number \( n \), expressed as \( \sqrt{n} \), is a value that, when multiplied by itself, gives \( n \).
For perfect squares such as 4, 9, and 16, the square roots are whole numbers because they revert back to their base number:

• \( \sqrt{4} = 2 \), because \( 2 \times 2 = 4 \).
• \( \sqrt{9} = 3 \), since \( 3 \times 3 = 9 \).
• \( \sqrt{16} = 4 \) because \( 4 \times 4 = 16 \).

Understanding square roots and being able to simplify them is crucial for effective algebraic manipulation, especially when dealing with expressions that involve radicals. Mastery of this concept is helpful not only for solving algebraic issues but also for various applications in geometry and calculus.