Problem 79
Question
Simplify each expression by performing the indicated operation. $$ \frac{4+\sqrt{5}}{4-\sqrt{5}} $$
Step-by-Step Solution
Verified Answer
Answer: The simplified form of the rational expression is $$\frac{21 + 8\sqrt{5}}{11}$$
1Step 1: Identify the given expression
The given expression is:
$$
\frac{4+\sqrt{5}}{4-\sqrt{5}}
$$
2Step 2: Rationalize the denominator
To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator, which is \((4+\sqrt{5})\):
$$
\frac{4+\sqrt{5}}{4-\sqrt{5}} \times \frac{4+\sqrt{5}}{4+\sqrt{5}}
$$
3Step 3: Multiply the numerators
Multiply the numerators as follows:
$$
(4+\sqrt{5})(4+\sqrt{5}) = 4^2 + 2(4)(\sqrt{5}) + (\sqrt{5})^2 = 16 + 8\sqrt{5} + 5
$$
4Step 4: Multiply the denominators
Multiply the denominators using the difference of squares formula:
$$
(4-\sqrt{5})(4+\sqrt{5}) = 4^2 - (\sqrt{5})^2 = 16 - 5
$$
5Step 5: Simplify the expression
Combine the results from Steps 3 and 4 to simplify the expression:
$$
\frac{16 + 8\sqrt{5} + 5}{16 - 5} = \frac{21 + 8\sqrt{5}}{11}
$$
The simplified expression is:
$$
\frac{21 + 8\sqrt{5}}{11}
$$
Key Concepts
Simplifying ExpressionsConjugate of a BinomialDifference of Squares
Simplifying Expressions
Simplifying mathematical expressions is a fundamental skill in algebra, involving the reduction of expressions to their simplest form. The goal is to make the expression as clear and concise as possible, often by combining like terms, eliminating complex fractions, or factoring.
For example, one common complication in expressions is the presence of radicals in the denominator. To simplify an expression like \( \frac{4+\sqrt{5}}{4-\sqrt{5}} \), it's necessary to rationalize the denominator, which means removing the radical to make it easier to work with. By following this process, you end up with a simpler expression that retains its original value but is more straightforward to use in calculations or further algebraic manipulation.
Simplifying expressions is an invaluable tool for solving equations, understanding functions, and analyzing mathematical relationships.
For example, one common complication in expressions is the presence of radicals in the denominator. To simplify an expression like \( \frac{4+\sqrt{5}}{4-\sqrt{5}} \), it's necessary to rationalize the denominator, which means removing the radical to make it easier to work with. By following this process, you end up with a simpler expression that retains its original value but is more straightforward to use in calculations or further algebraic manipulation.
Simplifying expressions is an invaluable tool for solving equations, understanding functions, and analyzing mathematical relationships.
Conjugate of a Binomial
The conjugate of a binomial plays a key role in the process of rationalization, especially when dealing with square roots. A binomial is a polynomial with two terms, such as \( 4 - \sqrt{5} \). The conjugate of this binomial is \( 4 + \sqrt{5} \), where the sign between the two terms is reversed.
When you multiply a binomial by its conjugate, the result is a special product known as the difference of squares, which will be explained in the next section. Utilizing the conjugate is a clever math trick that helps eliminate square roots from the denominator. By multiplying the numerator and the denominator of a fraction by the conjugate, you can drastically simplify the expression without changing its value.
When you multiply a binomial by its conjugate, the result is a special product known as the difference of squares, which will be explained in the next section. Utilizing the conjugate is a clever math trick that helps eliminate square roots from the denominator. By multiplying the numerator and the denominator of a fraction by the conjugate, you can drastically simplify the expression without changing its value.
Difference of Squares
The difference of squares is a pattern of algebra that occurs when you multiply a conjugate pair. Specifically, it states that for any two terms \( a \) and \( b \), the product of their difference and sum is equal to the difference of their squares: \( (a - b)(a + b) = a^2 - b^2 \). This formula is exceptionally useful in simplifying expressions.
In our example, when rationalizing the denominator, we apply the difference of squares to the denominator \( (4 - \sqrt{5})(4 + \sqrt{5}) \), resulting in \( 4^2 - (\sqrt{5})^2 \). Simplifying further yields \( 16 - 5 \), which is much more straightforward than dealing with a radical.
Understanding the difference of squares is not just about performing mechanical operations; it's about recognizing patterns that simplify complex algebraic expressions. This knowledge allows you to tackle higher-level math problems with confidence and efficiency.
In our example, when rationalizing the denominator, we apply the difference of squares to the denominator \( (4 - \sqrt{5})(4 + \sqrt{5}) \), resulting in \( 4^2 - (\sqrt{5})^2 \). Simplifying further yields \( 16 - 5 \), which is much more straightforward than dealing with a radical.
Understanding the difference of squares is not just about performing mechanical operations; it's about recognizing patterns that simplify complex algebraic expressions. This knowledge allows you to tackle higher-level math problems with confidence and efficiency.
Other exercises in this chapter
Problem 78
Find each of the following products. $$ \sqrt{5 x^{4} y^{3}}(\sqrt{8 x y}-5 \sqrt{7 x}) $$
View solution Problem 79
For the following problems, solve the equations. $$ \sqrt{2 x+5}=\sqrt{x+3} $$
View solution Problem 79
For the following problems, simplify each of the radical expressions. $$ \sqrt{(a-3)^{4}(a-1)^{2}} $$
View solution Problem 79
Simplify \(\sqrt{a^{2}+8 a+16}\).
View solution