Problem 30 - Algebra 2 and Trigonometry

Question

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers. \(\frac{7+\sqrt{5}}{7-\sqrt{5}}\)

Step-by-Step Solution

Answer

The simplest form is \(\frac{27}{22} + \frac{7\sqrt{5}}{22}\).

Step 1: Identify the Conjugate of the Denominator

To rationalize the denominator, we need to multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of a binomial of the form \(a - b\) is \(a + b\). Here, the denominator is \(7 - \sqrt{5}\), so its conjugate is \(7 + \sqrt{5}\).

Step 2: Multiply by the Conjugate

Multiply the entire fraction \(\frac{7+\sqrt{5}}{7-\sqrt{5}}\) by \(\frac{7+\sqrt{5}}{7+\sqrt{5}}\). This is equivalent to multiplying by 1, so it doesn't change the value of the fraction. The expression becomes: \[\frac{(7+\sqrt{5})(7+\sqrt{5})}{(7-\sqrt{5})(7+\sqrt{5})}\]

Step 3: Simplify the Numerator

Apply the distributive property (also known as FOIL for binomials) to expand the numerator:\[(7+\sqrt{5})(7+\sqrt{5}) = 7^2 + 7 \times \sqrt{5} + 7 \times \sqrt{5} + (\sqrt{5})^2 = 49 + 14\sqrt{5} + 5 = 54 + 14\sqrt{5}\]

Step 4: Simplify the Denominator

Similarly, apply the difference of squares formula to the denominator. The product of a binomial and its conjugate \((a-b)(a+b) = a^2 - b^2\) gives:\[(7-\sqrt{5})(7+\sqrt{5}) = 7^2 - (\sqrt{5})^2 = 49 - 5 = 44\]

Step 5: Write the Fraction in Simplest Form

Now, the expression simplifies to:\[\frac{54 + 14\sqrt{5}}{44}\]Each term in the numerator is divisible by 2, so we can simplify:\[\frac{54}{44} + \frac{14\sqrt{5}}{44} = \frac{27}{22} + \frac{7\sqrt{5}}{22}\]Thus, the fraction in its simplest form is:\[\frac{27}{22} + \frac{7\sqrt{5}}{22}\]

Key Concepts

Conjugates

Conjugates are essential when it comes to rationalizing denominators that contain irrational numbers, such as square roots. The conjugate of a binomial expression is generated by changing the sign between two terms. In the case of a binomial expressed as \(a - b\), the conjugate would be \(a + b\). Similarly, for \(a + b\), the conjugate is \(a - b\). This method is incredibly useful because it eliminates the radical part when multiplied with the original binomial, resulting in a rational number.
For instance, in the problem \(\frac{7 + \sqrt{5}}{7 - \sqrt{5}}\), the denominator is \(7 - \sqrt{5}\). Thus, its conjugate would be \(7 + \sqrt{5}\). By multiplying both the numerator and denominator by this conjugate, we get rid of the square root in the denominator. This operation forms a part of what's called the 'conjugate pairs', helping transform complicated fractions into simpler, more manageable expressions without changing their value. Remember, multiplying by a conjugate is like multiplying by 1, which is why the fraction's value stays the same.

Simplifying Fractions

Simplifying fractions involves reducing a fraction to its simplest or most compact form. This typically means finding the greatest common divisor (GCD) for the numerator and the denominator and then dividing both by the GCD. However, when dealing with fractions that involve radicals or roots, simplifying additionally involves ensuring that there is no irrational number in the denominator. This is often achieved by rationalizing the denominator.
In the exercise, once you've applied the conjugate, the expression \(\frac{54 + 14\sqrt{5}}{44}\) appears. This expression can be simplified further by noticing that both the numerator and denominator are divisible by common factors. For example, if each term in the numerator and the denominator can be divided by the same number, you perform the division to each term individually. Here, we divided by 2 to transform the fraction to its simplest form: \(\frac{27}{22} + \frac{7\sqrt{5}}{22}\). Throughout the process of simplifying, it is crucial not to change the fraction's actual value, only its form.

Difference of Squares

The difference of squares is a very helpful algebraic rule that simplifies the product of a binomial and its conjugate into a straightforward subtraction of squares:

• The formula for the difference of squares is \(a^2 - b^2\).
• It applies directly to the multiplication of conjugate pairs: \((a - b)(a + b) = a^2 - b^2\).

In the solution provided, when multiplying the conjugate \((7 + \sqrt{5})\) with the original denominator \((7 - \sqrt{5})\), this method drastically simplifies the calculation: \((7 - \sqrt{5})(7 + \sqrt{5}) = 7^2 - (\sqrt{5})^2 = 49 - 5 = 44\). This simplification occurs because the terms involving the square root cancel each other out, leaving a simple subtraction.
Recognizing situations to apply this concept can save a lot of time and effort. It is a frequently used technique in algebra to aid in rationalizing denominators and is key in advancing from basic algebra toward more complex mathematics.