Problem 66 - Algebra for College Students

Question

Use the distributive property to help simplify each of the following. All variables represent positive real numbers. \(-2 \sqrt{25 x}-4 \sqrt{36 x}+7 \sqrt{64 x}\)

Step-by-Step Solution

Answer

The simplified expression is \(22\sqrt{x}\).

Step 1: Simplify each square root

Begin by finding the square root for the perfect square inside each term. For \(\sqrt{25x}\), we have \(\sqrt{25} = 5\), so it becomes \(-2 \times 5 \times \sqrt{x} = -10\sqrt{x}\). Similarly, \(\sqrt{36x}\) becomes \(-4 \times 6 \times \sqrt{x} = -24\sqrt{x}\), and \(\sqrt{64x}\) becomes \(7 \times 8 \times \sqrt{x} = 56\sqrt{x}\).

Step 2: Combine like terms

Once each term is simplified, combine the like terms. We have the simplified terms: \(-10\sqrt{x} - 24\sqrt{x} + 56\sqrt{x}\).

Step 3: Perform the arithmetic

Add or subtract the coefficients of \(\sqrt{x}\). Perform \(-10 - 24 + 56\) which results in \(22\). So, we have \(22\sqrt{x}\).

Key Concepts

Simplifying Square Roots

In algebra, simplifying square roots is a crucial step to manage complex expressions. Let's break down the square roots in our problem step-by-step. Each term in our expression contains a square root of a product, like \(\sqrt{25x}\). To simplify such an expression, focus first on the perfect square part. This involves finding the integer whose square is equal to these numbers:

• \(\sqrt{25} = 5\) because \(5 \times 5 = 25\)
• \(\sqrt{36} = 6\) because \(6 \times 6 = 36\)
• \(\sqrt{64} = 8\) because \(8 \times 8 = 64\)

Once you've resolved the integer part of the square root, the product with the coefficient becomes straightforward. \(\sqrt{25x}\) transforms into \(5 \sqrt{x}\). With the original coefficients included, the term \(-2 \sqrt{25x}\) becomes \(-2 \times 5 \times \sqrt{x} = -10\sqrt{x}\), applying the same logic to the other terms.
Each simplified term retains the variable part unsimplified, such as \(\sqrt{x}\). This step prepares the terms for combining like terms later.

Combining Like Terms

Once all terms in an algebraic expression have been simplified, the next step is to combine like terms. Like terms are terms that have the exact same variable parts. For our example, all terms involve \(\sqrt{x}\), allowing us to combine them easily.

• From the preceding simplifications, we have: \(-10\sqrt{x}, -24\sqrt{x},\) and \(56\sqrt{x}\)

Combining these terms means adding or subtracting their coefficients while keeping the variable part \(\sqrt{x}\) constant. Conceptually, this is similar to consolidating similar items by adding or subtracting their amounts. Here, think about handling simple arithmetic with the coefficients:

• \(-10 - 24\) leads to \(-34\)
• Then, \(-34 + 56\) results in \(22\)

This gives us a single term of \(22\sqrt{x}\), considerably simplifying the expression. The combination of like terms streamlines multiple terms into a singular, more understandable mathematical phrase.

Arithmetic Operations with Radicals

Arithmetic operations with radicals follow similar rules to those applied in operations with whole numbers. However, due to the presence of square roots (radicals), attention is required to manage these components correctly. In our example, the key operation involved is arithmetic with like terms after simplification. Each simplified radical term has been reduced to a form that involves both a numerical coefficient and a radical, specifically \(\sqrt{x}\).
To perform arithmetic operations on these, you should:

• Treat the radicals as common variables if they are the same, in this case, \(\sqrt{x}\).
• Focus on the numeric coefficients that precede the radicals.
• Perform the arithmetic (addition or subtraction) on these coefficients.

In the given problem, once you've simplified each radical, combining them by subtracting and adding their coefficients logically follows: \(-10 - 24 + 56 = 22\).
This approach ensures that radicals with the same base \(\sqrt{x}\) are consolidated in a neat and orderly manner, highlighting the importance of managing coefficients along with radical operations.