Problem 6
Question
Fill in the blanks. The __________ of \(\sqrt{x}+1\) is \(\sqrt{x}-1\).
Step-by-Step Solution
Verified Answer
The conjugate of \( \sqrt{x} + 1 \) is \( \sqrt{x} - 1 \).
1Step 1: Understand the Concept of a Conjugate
The conjugate of a binomial is formed by changing the sign between two terms. If we have a binomial like \( a + b \), its conjugate would be \( a - b \). This is useful in algebra, especially when dealing with rationalizing denominators.
2Step 2: Identify the Terms in the Expression
The given expression is \( \sqrt{x} + 1 \). Here, the two terms are \( \sqrt{x} \) and \( 1 \). Notice that they are connected by a plus sign.
3Step 3: Formulate the Conjugate
To find the conjugate, replace the plus sign in \( \sqrt{x} + 1 \) with a minus sign. Thus, the conjugate of \( \sqrt{x} + 1 \) is \( \sqrt{x} - 1 \).
Key Concepts
Rationalizing DenominatorsAlgebraic ExpressionsBinomial Expressions
Rationalizing Denominators
Rationalizing denominators is a technique used to eliminate radicals from the denominators of algebraic expressions. This process can simplify mathematical expressions, making them easier to understand and solve. Imagine you have an expression with a denominator that includes a root, such as \( \frac{1}{\sqrt{2}} \). The challenge here is the presence of \( \sqrt{2} \) in the denominator.
To rationalize the denominator, you multiply both the numerator and denominator by the conjugate of the denominator. Let’s say your denominator is \( \sqrt{2} \). Since \( \sqrt{2} \) is a single term, its conjugate is technically itself. However, if it were a binomial like \( \sqrt{x} + 1 \), you would multiply by \( \sqrt{x} - 1 \). This multiplication removes the root, because \( ( \sqrt{x} + 1 )( \sqrt{x} - 1 ) = x - 1 \).
This approach leverages the difference of squares, where \( (a + b)(a - b) = a^2 - b^2 \). For \( x = (\sqrt{2})^2 \), you have a denominator of 2 after multiplication.
To rationalize the denominator, you multiply both the numerator and denominator by the conjugate of the denominator. Let’s say your denominator is \( \sqrt{2} \). Since \( \sqrt{2} \) is a single term, its conjugate is technically itself. However, if it were a binomial like \( \sqrt{x} + 1 \), you would multiply by \( \sqrt{x} - 1 \). This multiplication removes the root, because \( ( \sqrt{x} + 1 )( \sqrt{x} - 1 ) = x - 1 \).
This approach leverages the difference of squares, where \( (a + b)(a - b) = a^2 - b^2 \). For \( x = (\sqrt{2})^2 \), you have a denominator of 2 after multiplication.
- Multiplying by conjugates is key to rationalizing.
- This process leverages difference of squares.
- It simplifies numerical evaluation.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. They can be as simple as \( x + 2 \) or more complex like \( x^2 + 3x + 2 \). Understanding algebraic expressions is fundamental in algebra because they represent equations and functions and are used to solve problems.
An algebraic expression can include:
Recognizing how components in algebraic expressions relate and manipulating them appropriately is crucial for solving algebraic equations, making this an indispensable tool for students.
An algebraic expression can include:
- Constants, which are numbers on their own (like 2 or 5).
- Variables, which can take various values (like \( x \) or \( y \)).
- Operations, which include addition, subtraction, multiplication, etc.
Recognizing how components in algebraic expressions relate and manipulating them appropriately is crucial for solving algebraic equations, making this an indispensable tool for students.
Binomial Expressions
A binomial expression consists of two terms separated by a plus or minus sign, such as \( a + b \) or \( a - b \). Binomials are specific types of polynomial expressions, and understanding them is essential when dealing with algebraic concepts like conjugates or factoring.
The term "conjugate," especially in the context of binomials, involves creating a new expression by changing the sign between the two terms. The conjugate of \( \sqrt{x} + 1 \) is \( \sqrt{x} - 1 \). This is particularly useful when eliminating radicals from denominators, as already shown.
Key applications of binomial expressions include:
The term "conjugate," especially in the context of binomials, involves creating a new expression by changing the sign between the two terms. The conjugate of \( \sqrt{x} + 1 \) is \( \sqrt{x} - 1 \). This is particularly useful when eliminating radicals from denominators, as already shown.
Key applications of binomial expressions include:
- In algebraic equations, helping solve quadratic equations.
- Facilitation of certain integrals or derivatives in calculus.
- Simplifying complex numbers in the form of \( a + bi \).
Other exercises in this chapter
Problem 6
Fill in the blanks. \(32^{4 / 5}\) means the fourth _____ of the fifth _____ of 32.
View solution Problem 6
Fill in the blanks. Simplify: $$ \sqrt{-36}=\sqrt{\cdot 36}=\sqrt{36}=6 $$
View solution Problem 6
Fill in the blanks. In any right triangle, the square of the hypotenuse is equal to the _____ of the squares of the two ____.
View solution Problem 6
Fill in the blanks. The quotient rule for radicals: \(\sqrt[n]{\frac{a}{b}}=\). In words, the root of the ____ of two numbers is equal to the quotient of their
View solution