Problem 39 - Intermediate Algebra

Question

Write the conjugate of each expression. $$ -7 \sqrt{5}+8 \sqrt{x} $$

Step-by-Step Solution

Answer

The conjugate is \(-7 \sqrt{5} - 8 \sqrt{x}\).

Step 1: Identify the Given Expression

The given expression is \(-7 \sqrt{5} + 8 \sqrt{x}\). This is a binomial expression containing two terms.

Step 2: Understand the Concept of a Conjugate

The conjugate of a binomial expression of the form \(a \pm b\) is obtained by switching the plus or minus sign between the terms. For instance, the conjugate of \(a + b\) is \(a - b\), and vice versa.

Step 3: Determine the Conjugate

In the expression \(-7 \sqrt{5} + 8 \sqrt{x}\), switch the plus sign to a minus sign to find its conjugate. The conjugate is \(-7 \sqrt{5} - 8 \sqrt{x}\).

Key Concepts

Binomial Expressions

A binomial expression consists of exactly two distinct terms. In the context of algebra, these terms are often variables, constants, or a combination involving operations such as addition or subtraction. The expression that was part of the original exercise, \(-7 \sqrt{5} + 8 \sqrt{x}\), is a perfect example of a binomial. Here, we have two terms connected by a plus sign:
1. \(-7 \sqrt{5}\)
2. \(8 \sqrt{x}\)

Each part of a binomial can have different mathematical properties or forms, such as containing radicals or simple numbers. Binomials are essential in algebra because they allow us to illustrate basic polynomial operations and to explore more complex algebraic structures as we learn to manipulate them.
When dealing with binomials, we often look for patterns like the conjugate, expand expressions using FOIL (First, Outer, Inner, Last), or factor them into products to simplify or solve equations.

Radicals in Algebra

Radicals are symbols used in algebra to denote roots, such as square roots or cube roots. They involve expressions within the radical sign, which need to be simplified or manipulated according to algebraic rules. In the exercise expression \(-7 \sqrt{5} + 8 \sqrt{x}\), both terms include radicals, \(\sqrt{5}\) and \(\sqrt{x}\).

Understanding radicals includes knowing how to:

• Simplify radicals by factoring out squares, cubes, or other powers.
• Manipulate radical expressions, such as multiplying and dividing these terms.
• Rationalize denominators, which further allows the expression to be simplified.

Radicals often appear in binomial or polynomial expressions, adding layers of complexity that are fundamental to grasping advanced algebra. By familiarizing oneself with the basic principles and rules surrounding radicals, solving and simplifying expressions becomes more straightforward. This is critical to tackling problems involving binomials like conjugates.

Algebraic Expressions

The backbone of algebra is algebraic expressions, which are combinations of numbers, variables, and operations. These expressions serve as the foundation for equations, which are statements that set two expressions equal. In this context, the given expression \(-7 \sqrt{5} + 8 \sqrt{x}\) falls under the category of algebraic expressions due to its use of radicals and operations.

Algebraic expressions can range from simple sums and products to complex polynomials involving multiple variables. They are manipulated by:

• Adding or subtracting like terms.
• Distributing variables or constants across terms.
• Applying rules for exponents and radicals.

Mastering the manipulation of algebraic expressions leads to solving equations and inequalities, which is a critical skill in mathematics. By thoroughly understanding how to handle expressions of all types and forms, including binomials and radicals, students build a robust foundation for tackling abstract mathematical concepts.