Problem 29 - College Algebra

Question

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

The denominator of the expression is \(1 - \sqrt{17}\). In order to rationalize the denominator, we use its conjugate, which is \(1 + \sqrt{17}\).

Step 2: Multiply by the Conjugate

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

Step 3: Expand the Denominator

The denominator becomes: \[(1-\sqrt{17})(1+\sqrt{17}) = 1^2 - (\sqrt{17})^2 = 1 - 17 = -16\]

Step 4: Expand the Numerator

The numerator becomes: \[8(1+\sqrt{17}) = 8 + 8\sqrt{17}\]

Step 5: Simplify the Expression

Combine the simplified numerator and denominator to get: \[\frac{8 + 8\sqrt{17}}{-16} = -\frac{8}{16} - \frac{8\sqrt{17}}{16} = -\frac{1}{2} - \frac{\sqrt{17}}{2}\]

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators such as addition, subtraction, multiplication, and division. They can sometimes look intimidating, especially when they include square roots or other complex terms. The key to dealing with algebraic expressions is to understand how to manipulate them using standard rules of arithmetic and algebra.

• Numerical terms involved in expressions usually include integers or fractions.
• Variable terms represent unknown values and are depicted by symbols like \(x\), \(y\), or \(z\).
• Operators join numerical and variable components to build formulas or expressions.

Breaking down complex expressions into smaller, more manageable parts can make working with them easier. Understanding how each part functions helps in performing more complex operations like simplifying or rationalizing expressions.

Conjugates

Conjugates come into play to help us eliminate irrational numbers from the denominator, making expressions easier to work with. In mathematics, the conjugate of a binomial expression like \(a - b\) is \(a + b\) and vice versa.

• When multiplied, conjugates help in removing the square root from an expression because \((a - b)(a + b) = a^2 - b^2\).
• This transformation takes something potentially cumbersome and simplifies it, paving the way for more straightforward manipulation.

In the context of rationalizing denominators, conjugates are most beneficial. For example, in the expression \(\frac{8}{1-\sqrt{17}}\), the conjugate \(1 + \sqrt{17}\) helps simplify it by forming a difference of squares in the denominator: \(1^2 - (\sqrt{17})^2 = 1 - 17 = -16\).This results in reducing the complexity of the denominator, making the overall expression simpler.

Simplifying Expressions

Simplifying expressions is all about making them more concise and easier to interpret. The goal is to turn a complex expression into its simplest form without changing its value. During simplification, you typically perform operations such as expanding, combining like terms, and reducing fractions.

• Expanding involves distributing numbers across parentheses, as seen when transforming \(8(1+\sqrt{17})\) into \(8 + 8\sqrt{17}\).
• Combining like terms allows us to join terms that are similar, reducing their number and clarifying the expression.
• Reducing fractions involves dividing both the numerator and the denominator by their greatest common factor, simplifying them as much as possible.

In the exercise, simplifying the expression from \(\frac{8 + 8\sqrt{17}}{-16}\) to \(-\frac{1}{2} - \frac{\sqrt{17}}{2}\) shows how numerical coefficients and like terms can work together. This transformation provides a cleaner, easier format, making further calculations or analyses more straightforward.