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Magazine Chapter 9: Problem 40
For the following problems, simplify the expressions. $$ \frac{4+\sqrt{11}}{4-\sqrt{11}} $$
Short Answer
Expert verified
Question: Simplify the following expression:
$$
\frac{4 + \sqrt{11}}{4 - \sqrt{11}}
$$
Answer:
$$
\frac{27 + 8\sqrt{11}}{5}
$$
Step by step solution
01
Identify the conjugate
The conjugate of a binomial expression is obtained by changing the sign of the second term. In this case, the conjugate of the denominator, 4 - sqrt(11), would be 4 + sqrt(11).
02
Multiply the numerator and the denominator by the conjugate
Multiply both the numerator and the denominator of the fraction by the conjugate, 4 + sqrt(11), to eliminate the square root from the denominator:
$$
\frac{4 + \sqrt{11}}{4 - \sqrt{11}} \times \frac{4 + \sqrt{11}}{4 + \sqrt{11}}
$$
03
Multiply the numerators and denominators
Multiply the numerators and the denominators separately. Use the distributive property (FOIL) to expand the expressions:
$$
\frac{(4 + \sqrt{11})(4 + \sqrt{11})}{(4 - \sqrt{11})(4 + \sqrt{11})}
$$
04
Simplify the numerator and the denominator
Simplify the numerator and the denominator by applying the difference of squares formula:
$$
\frac{4^2 + 2 \cdot 4\sqrt{11} + (\sqrt{11})^2}{4^2 - (\sqrt{11})^2}
$$
05
Calculate the result
Now, calculate the result by substituting the values:
$$
\frac{16 + 8\sqrt{11} + 11}{16 - 11}
$$
06
Simplify the fraction
Combine the like terms in the numerator and simplify the denominator:
$$
\frac{27 + 8\sqrt{11}}{5}
$$
The simplified expression is:
$$
\frac{27 + 8\sqrt{11}}{5}
$$
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Conjugate Method
The conjugate method is a technique used in algebra to simplify expressions, especially when dealing with radicals. A conjugate is formed by changing the sign between two terms of a binomial. For instance, if you have an expression like \(4 - \sqrt{11}\), its conjugate would be \(4 + \sqrt{11}\). The main goal of the conjugate method is to remove irrational numbers, like square roots, from the denominator of a fraction.
- This is achieved by multiplying both the numerator and the denominator by the conjugate of the denominator.
- This helps to create a difference of squares, which leads to easy simplification.
Rationalizing the Denominator
Rationalizing the denominator means converting a fraction with an irrational denominator into an equivalent fraction with a rational denominator. This process simplifies expressions and makes calculations easier. Imagine you want to simplify \( \frac{4+\sqrt{11}}{4-\sqrt{11}} \).
- First, identify the conjugate of the denominator.
- Then, multiply both the numerator and the denominator by this conjugate.
Distributive Property (FOIL)
The distributive property is fundamental in multiplying binomials. Specifically, the FOIL method is a mnemonic to help remember the steps for multiplying two binomials: First, Outer, Inner, Last.
For instance, multiplying \((4 + \sqrt{11})(4 + \sqrt{11})\) involves:
For instance, multiplying \((4 + \sqrt{11})(4 + \sqrt{11})\) involves:
- First: Multiply the first terms: \(4 \times 4 = 16\)
- Outer: Multiply the outer terms: \(4 \times \sqrt{11} = 4\sqrt{11}\)
- Inner: Multiply the inner terms: \(\sqrt{11} \times 4 = 4\sqrt{11}\)
- Last: Multiply the last terms: \(\sqrt{11} \times \sqrt{11} = 11\)
Difference of Squares Formula
The difference of squares formula is a powerful tool when simplifying expressions involving squares. It's expressed as \((a + b)(a - b) = a^2 - b^2\). In the context of rationalizing denominators, like in our problem, it simplifies multiplication by using the formula directly.
Applying the formula to \((4 - \sqrt{11})(4 + \sqrt{11})\), you get:
Applying the formula to \((4 - \sqrt{11})(4 + \sqrt{11})\), you get:
- \(a = 4\) and \(b = \sqrt{11}\)
- Substitute into the formula: \(4^2 - (\sqrt{11})^2\)
- This yields \(16 - 11 = 5\)
