Problem 33 - College Algebra

Question

In Exercises \(27-38,\) add or subtract terms whenever possible. $$\sqrt{50 x}-\sqrt{8 x}$$

Step-by-Step Solution

Answer

The simplified result of \(\sqrt{50x} - \sqrt{8x}\) is \(3 \sqrt{2x}\).

Step 1: Simplify the square root terms

It is noticeable that both terms contain square roots, which can be simplified by factoring out perfect squares. The term \(\sqrt{50x}\) can be factored out into \(\sqrt{25 \cdot 2x}\) which simplifies into \(5 \sqrt{2x}\). Similarly, the term \(\sqrt{8x}\) can be factored out into \(\sqrt{4 \cdot 2x}\) which simplifies into \(2 \sqrt{2x}\).

Step 2: Evaluate the expression

Now that both terms are simplified, we substitute them back into the original expression: So, \( \sqrt{50x} - \sqrt{8x}\) becomes \(5 \sqrt{2x} - 2 \sqrt{2x}\). This expression can now be further simplified, since both terms contain the common factor \( \sqrt{2x}\).

Step 3: Simplify the final expression

Since both terms contain a common factor, we perform the subtraction operation on the coefficients, resulting in the expression: \(5 \sqrt{2x} - 2 \sqrt{2x}= 3 \sqrt{2x}\).

Key Concepts

Square Root Simplification

Square root simplification is all about breaking down numbers into their simplest radical form. When simplifying square roots, the goal is to find the largest perfect square that can divide into the number evenly. A perfect square is a whole number that can be expressed as the square of an integer, like 1, 4, 9, 16, and so on. Let's break it down:

• Look for perfect square factors: For example, in the term \( \sqrt{50x} \), 25 is a perfect square that divides into 50, which gives \( \sqrt{25 \cdot 2x} \).
• Take the square root of the perfect square: The square root of 25 is 5, simplifying \( \sqrt{25 \cdot 2x} \) to \( 5\sqrt{2x} \).

In this way, we've simplified \( \sqrt{50x} \) to \( 5\sqrt{2x} \). Similarly, \( \sqrt{8x} \) was reduced to \( 2\sqrt{2x} \). Simplifying radicals helps in performing arithmetic operations more efficiently.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables (like \( x \)), and operation symbols. Understanding algebraic expressions is key to simplifying and manipulating mathematical problems. Here's a closer look at how this works:

• Identify like terms: In expressions such as \( 5\sqrt{2x} \) and \( 2\sqrt{2x} \), like terms are those that have the same variable part—in this case, \( \sqrt{2x} \).
• Combine coefficients: Once like terms are identified, you can simply add or subtract their coefficients. So, \( 5\sqrt{2x} - 2\sqrt{2x} \) becomes \( (5 - 2)\sqrt{2x} \).

Recognizing and combining like terms makes calculations simpler and expressions clearer to understand. Algebraic expressions lay the foundation for solving equations and inequalities.

Factoring Perfect Squares

Factoring perfect squares involves expressing a number or expression as a product of perfect squares. It is a crucial step in simplifying radicals and can make seemingly complex expressions much more manageable. Here's how it works:

• Find the greatest perfect square factor: For instance, in \( \sqrt{50x} \), identify 25 as the perfect square, since \( 25 = 5^2 \).
• Rearrange the expression: Rewrite the expression as a product involving this perfect square, like \( \sqrt{25 \cdot 2x} \).
• Apply the square root: Replace the perfect square with its root to simplify the expression. This results in \( 5\sqrt{2x} \) from \( \sqrt{25 \cdot 2x} \).

Factoring perfect squares simplifies computations and is an essential skill for tackling more advanced algebraic problems. Practice helps in quickly spotting and factoring perfect squares within expressions.