Problem 37 - Intermediate Algebra

Question

Write the conjugate of each expression. $$ 5-\sqrt{a} $$

Step-by-Step Solution

Answer

The conjugate is \( 5 + \sqrt{a} \).

Step 1: Understanding the Concept

The conjugate of a binomial expression in the form of \( a - \sqrt{b} \) is \( a + \sqrt{b} \). The conjugate involves simply changing the sign between the two terms of the binomial.

Step 2: Identifying Components

Identify the components of the given expression \( 5 - \sqrt{a} \). Here, \( a = 5 \) and \( b = a \) in the binomial expression form \( a - \sqrt{b} \).

Step 3: Applying the Conjugate Rule

Using the concept from Step 1, apply the conjugate rule to change the sign in the expression \( 5 - \sqrt{a} \). This results in the expression \( 5 + \sqrt{a} \).

Key Concepts

Binomial Expression

A binomial expression is an algebraic expression containing exactly two terms. These terms are typically joined by either a plus or a minus sign. In algebra,

• The structure of a binomial expression can be represented as \( a + b \) or \( a - b \).
• Such expressions are common in problems involving conjugates, particularly those dealing with square roots.

For instance, in the problem involving the expression \( 5 - \sqrt{a} \), we have a binomial with two terms: 5 and \( -\sqrt{a} \). Here, recognizing it as a binomial is crucial for performing operations such as finding the conjugate.

Square Roots

Square roots appear often in algebraic expressions, especially when dealing with binomials. A square root is the value that, when multiplied by itself, gives the original number. For example:

• If \( x^2 = a \), then \( x = \sqrt{a} \).
• In expressions like \( \sqrt{a} \), \( a \) is under the square root symbol, which indicates the operation of finding a number which squared equals \( a \).

In our specific expression \( 5 - \sqrt{a} \), the term \( \sqrt{a} \) represents the square root component of the binomial. Understanding how square roots interact with other elements in an expression is key for tasks like determining conjugates.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying algebraic expressions through various operations. When working with binomials and square roots, such as in the context of finding a conjugate, this manipulation often involves changing signs and rearranging terms.To find the conjugate:

• Identify your binomial expression, which in our example is \( 5 - \sqrt{a} \).
• The conjugate is formed by switching the sign between the terms, resulting in \( 5 + \sqrt{a} \).

Such manipulation helps in rationalizing denominators or simplifying expressions, especially when tackling algebra problems involving irrational numbers like square roots. Mastering these concepts can make algebra more approachable and easier to navigate.