Problem 29 - College Algebra

Question

For the following exercises, simplify each expression. $$ \frac{8}{1-\sqrt{17}} $$

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{1}{2} - \frac{\sqrt{17}}{2}.\)

Step 1: Identify the Conjugate

To simplify the expression \( \frac{8}{1-\sqrt{17}} \), recognize that the denominator has a radical. To eliminate the radical, multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(1 - \sqrt{17}\) is \(1 + \sqrt{17}\).

Step 2: Multiply by the Conjugate

Multiply both the numerator and denominator by \(1 + \sqrt{17}\): \[\frac{8}{1-\sqrt{17}} \cdot \frac{1+\sqrt{17}}{1+\sqrt{17}} = \frac{8(1+\sqrt{17})}{(1-\sqrt{17})(1+\sqrt{17})}.\]

Step 3: Simplify the Denominator

Use the formula \((a-b)(a+b) = a^2 - b^2\) to simplify the denominator: \[(1-\sqrt{17})(1+\sqrt{17}) = 1^2 - (\sqrt{17})^2 = 1 - 17 = -16.\]

Step 4: Simplify the Numerator

Distribute \(8\) in the numerator:\[ 8(1+\sqrt{17}) = 8 + 8\sqrt{17}.\]

Step 5: Combine and Simplify the Expression

Combine the results from the steps above to write the simplified expression:\[\frac{8 + 8\sqrt{17}}{-16} = \frac{8}{-16} + \frac{8\sqrt{17}}{-16} = -\frac{1}{2} - \frac{\sqrt{17}}{2}.\]

Step 6: Write the Final Simplified Expression

The final simplified expression is: \(-\frac{1}{2} - \frac{\sqrt{17}}{2}.\)

Key Concepts

Conjugate

In mathematics, the conjugate of a binomial expression is an important concept, especially when dealing with radicals in the denominator. A binomial expression refers to an expression with two terms, such as \(1 - \sqrt{17}\). The conjugate simply involves flipping the sign between these terms. Hence, the conjugate of \(1 - \sqrt{17}\) is \(1 + \sqrt{17}\).

• If the original expression is \(a - b\), then its conjugate is \(a + b\), and vice-versa.
• Conjugates are very useful because they help eliminate radicals or irrational numbers from denominators.

When you multiply a binomial by its conjugate, you make use of the difference of squares formula: \((a-b)(a+b) = a^2 - b^2\). In the step-by-step problem, multiplying \((1 - \sqrt{17})(1 + \sqrt{17})\) uses this formula to yield an integer, thereby removing the radical from the denominator. By understanding conjugates, you can rationalize denominators and express fractions in a simpler form.

Radical Expressions

Radical expressions include roots such as square roots, cube roots, etc. The expression \(\sqrt{17}\) is an example of a radical expression. In algebra, sometimes it's necessary to manipulate these expressions to make them easier to work with, usually by removing radicals from the denominators.

Manipulating radical expressions often involves:

• Identifying and using conjugates, as discussed in the previous section.
• Simplifying the radicand, which is the value under the radical sign, if possible.

For example, in the original exercise, the fraction \(\frac{8}{1-\sqrt{17}}\) presents a radical in the denominator, which can make calculations and further manipulations complex. Through rationalizing the denominator using the conjugate, you transform the expression into an equivalent one with a simpler format, ultimately eliminating the radical in the denominator.

This makes the fraction easier to understand and work with, especially in more complex algebraic computations.

Simplifying Algebraic Fractions

Simplifying algebraic fractions is key to making expressions more manageable. This involves combining like terms, reducing fractions to their simplest form, and ensuring there's no unnecessary complexity, like radicals in the denominator.

When simplifying a fraction like \(\frac{8 + 8\sqrt{17}}{-16}\), you need to apply basic fraction and algebraic simplification principles:

• Factor out common terms in the numerator and denominator and cancel them out, if possible.
• Break down combined terms into simpler fractions, as seen when rewriting \(\frac{8}{-16} + \frac{8\sqrt{17}}{-16}\).
• Finally, simplify each part to its lowest terms, resulting in a clean expression.

Through these steps in the solution, you see how the original expression becomes \(-\frac{1}{2} - \frac{\sqrt{17}}{2}\), which is much simpler and often preferred in mathematical computations. Simplifying algebraic fractions ensures clarity and ease in handling further operations, making mathematical expressions not only concise but more elegant.