Problem 35 - Algebra and Trigonometry

Question

Add or subtract terms whenever possible. $$ 6 \sqrt{17 x}-8 \sqrt{17 x} $$

Step-by-Step Solution

Answer

The simplified form of the expression is \(14\sqrt{17 x}\)

Step 1: Identify like terms

In this expression, the like terms are \(6 \sqrt{17 x}\) and \(-8 \sqrt{17 x}\). The like parts of these terms are the \(\sqrt{17 x}\). So these terms can be combined through addition or subtraction.

Step 2: Combine like terms

To combine the like terms, use arithmetic operations on their coefficients. In this case, subtract the coefficient of the second term from the coefficient of the first term: \(6 - (-8) = 14\). Now the expression simplifies to \(14\sqrt{17 x}\)

Key Concepts

Combining like terms

Combining like terms is an essential skill in algebra that helps simplify expressions. In mathematics, terms are considered 'like' if they contain the same variables raised to the same power, differing only in their coefficients. For example, variables like \( \sqrt{17x} \) in the expression \( 6\sqrt{17x} - 8\sqrt{17x} \) are considered like terms because they carry the same radical and variable parts, differing only by their coefficients (6 and -8 respectively). The aim is to perform arithmetic operations on these coefficients while keeping the variable part unchanged.

When combining like terms, the process is straightforward:

• Identify the like terms.
• Take note of their coefficients.
• Use basic arithmetic operations (addition or subtraction) on these coefficients.

In our example, the coefficients 6 and -8 are combined by subtraction to simplify the expression, resulting in \( -2\sqrt{17x} \). This reduces complexity and helps in further mathematical operations.

Simplifying expressions

Simplifying expressions is all about making mathematical phrases easier to understand or work with, while ensuring the expression remains equivalent to the original. This process often involves several techniques such as combining like terms, factoring, or reducing fractions. The most common step in simplifying is combining like terms, as seen in the initial example.

To simplify an expression like \( 6\sqrt{17x} - 8\sqrt{17x} \), we:

• Identify like terms, which here means noticing they both have the radical \( \sqrt{17x} \).
• Subtract the coefficients (6 and 8) to find the new coefficient (-2).
• Reattach the common radical to the simplified coefficient, leading to \( -2\sqrt{17x} \).

These steps reduce the expression to a simple form, making it more manageable for further calculations or analyses. Effective expression simplification can transform algebraic challenges into more accessible tasks, often revealing patterns or solutions.

Algebraic expressions

An algebraic expression is a combination of numbers, variables, and operations (such as addition, subtraction, multiplication, and sometimes division). They form the backbone of algebra and can represent real-world phenomena in mathematical form. Each part of an algebraic expression, such as constants (fixed numbers) and variables (symbols standing in for numbers), plays a crucial role.

Consider the expression \( 6\sqrt{17x} - 8\sqrt{17x} \) from our example:

• Here, \( \sqrt{17x} \) includes both a constant (17) inside the radical and a variable (x).
• The operation (\