Problem 40 - Intermediate Algebra

Question

Write the conjugate of each expression. $$ -9 \sqrt{2}-6 \sqrt{y} $$

Step-by-Step Solution

Answer

The conjugate is \(-9\sqrt{2} + 6\sqrt{y}\).

Step 1: Understand the Concept of Conjugates

The conjugate of a binomial expression in the form \(a + b\) or \(a - b\) is found by changing the sign between the two terms. This means if we have \(a + b\), its conjugate is \(a - b\), and vice versa.

Step 2: Identify the Expression Type

The given expression is \(-9\sqrt{2} - 6\sqrt{y}\), which is in the form \(a - b\). Here, \(a = -9\sqrt{2}\) and \(b = 6\sqrt{y}\).

Step 3: Write the Conjugate

To find the conjugate of \(-9\sqrt{2} - 6\sqrt{y}\), change the sign between the two terms: \(-9\sqrt{2} + 6\sqrt{y}\).

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics. They represent a number that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3 because 3 multiplied by 3 equals 9. In algebraic expressions, square roots are denoted with the radical symbol, \(\sqrt{ }\).

Let's consider \(\sqrt{y}\), an example of a variable under a square root. When dealing with square roots in algebra, simplifying expressions often involves rationalizing the denominator or finding conjugates.

• Rationalizing involves eliminating the square root from the denominator of a fraction.
• Conjugates are useful when combining terms with square roots, especially in sums or differences.

Understanding square roots can demystify how they interact with algebraic expressions, leading to simplified or more manageable forms.

Binomial Expressions

Binomial expressions are algebraic expressions that consist of exactly two terms. These expressions can take forms such as \(a + b\) or \(a - b\). In our exercise, the given expression is \(-9\sqrt{2} - 6\sqrt{y}\), a typical binomial form.

When dealing with binomials, one common operation is finding their conjugates. Conjugates are especially important when the binomials involve square roots, as they can be used to eliminate square roots during multiplication. For instance:

• The conjugate of \(a + b\) is \(a - b\).
• The conjugate of \(a - b\) is \(a + b\).

These properties are employed to simplify expressions and solve equations, making binomials a central topic in algebra.

Algebraic Expressions

Algebraic expressions are combinations of variables, constants, and arithmetic operations. They form the building blocks of algebra and can range from simple expressions like \(x + 2\) to more complex ones with multiple terms and operations.

In the context of this exercise, understanding algebraic expressions includes identifying types like binomials and manipulating them using operations such as simplification or finding conjugates. The given expression \(-9\sqrt{2} - 6\sqrt{y}\) is an algebraic expression that involves both constants and terms with square roots.

Key skills when working with algebraic expressions include:

• Adding and subtracting terms.
• Simplifying expressions by combining like terms or using properties of numbers.
• Performing operations like finding conjugates to simplify or solve equations.

These abilities enable students to break down complex problems into manageable parts, leading to solutions that are easier to understand and verify.