Problem 35 - College Algebra Essentials

Question

$$6 \sqrt{17 x}-8 \sqrt{17 x}$$

Step-by-Step Solution

Answer

\(-2 \sqrt{17x}\)

Step 1: Recognize like terms

You have 6 times the square root of 17x and 8 times the square root of 17x. As the base (in this case, 17x under the root operation) is the same, these are like terms.

Step 2: Perform Subtraction

Subtract the second term from the first: \(6 \sqrt{17x} - 8 \sqrt{17x}\). This is akin to normal subtraction of terms, like if you had \(6x - 8x\)

Step 3: Simplify Result

After you subtract, you get: \(-2 \sqrt{17x}\). As a rule, when you perform addition or subtraction between like terms, the variable part (or, in this case, the radical part, remains the same.

Key Concepts

Like Terms

In mathematics, the concept of like terms is vital when you're dealing with algebraic expressions, especially when simplifying. Like terms are terms that have the exact same variables raised to the same powers. In the context of radicals, like terms have the same radicand (the number under the root). So, in our original exercise, both terms are like terms because they have the same radicand, \(\sqrt{17x}\). These terms can be combined just like you would combine any other algebraic expressions with the same variables. It’s similar to combining \(6x\) and \(8x\) because they both have an \(x\). Recognizing this allows us to add or subtract these terms much like simple numbers.

Subtraction of Radicals

Subtracting radicals can be simplified by focusing on their like terms. In this case, the expression \(6 \sqrt{17x} - 8 \sqrt{17x}\), involves subtracting like radicals.

The trick here is to treat the radicals as if they were an algebraic term such as \(x\). Suppose you have \(6x - 8x\). You would subtract the coefficients: \(6 - 8\) to get \(-2x\). Similarly, with radicals, you subtract the coefficients of \(\sqrt{17x}\).

Here’s how it looks:

• Subtract the coefficients: 6 - 8 = -2.
• Attach the unchanged radical part: \(\sqrt{17x}\).

So, the result becomes \(-2 \sqrt{17x}\). Notice how the subtraction process does not alter the radical part, only the number before it.

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form. With radicals, this means ensuring the expression is reduced as much as possible, but without losing its equivalent value.

In our exercise, after recognizing the like terms and performing subtraction, we reached \(-2 \sqrt{17x}\). This is a simplified form because it cannot be reduced further without changing the expression's value or meaning.

Simplifying radicals primarily focuses on:

• Identifying and combining like terms.
• Performing arithmetic operations on coefficients.
• Ensuring the radicand does not have perfect squares (other than 1) left.

Remember, a fully simplified radical expression is essential for clarity and ease in problem-solving across all mathematics levels.