Problem 39 - Algebra A Combined Function

Question

Rationalize each denominator. Assume that all variables represent positive real numbers. \(\frac{8}{1+\sqrt{10}}\)

Step-by-Step Solution

Answer

The rationalized form is \(-\frac{8}{9} + \frac{8\sqrt{10}}{9}\).

Step 1: Identify the Conjugate

To rationalize the denominator of the expression \(\frac{8}{1+\sqrt{10}}\), we need to use the conjugate of the denominator. The conjugate of \(1+\sqrt{10}\) is \(1-\sqrt{10}\).

Step 2: Multiply Numerator and Denominator by the Conjugate

Multiply the numerator and the denominator by the conjugate \(1-\sqrt{10}\) to get:\[ \frac{8}{1+\sqrt{10}} \times \frac{1-\sqrt{10}}{1-\sqrt{10}} = \frac{8(1-\sqrt{10})}{(1+\sqrt{10})(1-\sqrt{10})} \]

Step 3: Simplify the Denominator

The denominator is a difference of squares, so simplify it: \[(1+\sqrt{10})(1-\sqrt{10}) = 1^2 - (\sqrt{10})^2 = 1 - 10 = -9\]

Step 4: Distribute in the Numerator

Distribute in the numerator to simplify the expression: \[ 8(1-\sqrt{10}) = 8 \cdot 1 - 8 \cdot \sqrt{10} = 8 - 8\sqrt{10} \]

Step 5: Write the Simplified Expression

Combine the simplified numerator and the simplified denominator to write the result:\[ \frac{8 - 8\sqrt{10}}{-9} = -\frac{8}{9} + \frac{8\sqrt{10}}{9} \]

Step 6: State the Final Simplified Expression

The expression with the rationalized denominator is: \[ -\frac{8}{9} + \frac{8\sqrt{10}}{9} \]

Key Concepts

Conjugate Pairs

When rationalizing denominators, one key technique involves using conjugate pairs. These pairs are two expressions that are identical except for the sign between two terms. For example, in the expression given by the problem, the denominator is \(1+\sqrt{10}\). Its conjugate is \(1-\sqrt{10}\). By multiplying a fraction by a conjugate, you can simplify it, particularly when facing square roots.

• Conjugates typically involve square roots; one term's sign is reversed.
• This process eliminates radicals in the denominator.
• To rationalize denominators, apply conjugates effectively.

With conjugates, the key is to multiply both the numerator and denominator of the fraction by the conjugate form of the denominator. This method not only maintains the value of the original expression but transforms the denominator into an easier form to handle.

Difference of Squares

The `difference of squares` is a crucial algebraic concept used when dealing with conjugates. When you multiply two conjugate expressions, like \((1+\sqrt{10})(1-\sqrt{10})\), the terms in the center cancel each other out. This simplifies the expression to a form known as the difference of squares.

• A difference of squares formula: \((a+b)(a-b) = a^2 - b^2\).
• It results in a radical-free expression or integer.
• This method greatly simplifies mathematical expressions.

In the original exercise, the difference of squares simplifies \((1+\sqrt{10})(1-\sqrt{10})\) to \[1^2 - (\sqrt{10})^2 = 1 - 10 = -9\]. This integer denominator is much simpler for further arithmetic operations, allowing the overall fraction to be simplified further.

Simplifying Fractions

Simplifying fractions is a fundamental skill in algebra. The goal is to rewrite fractions in their simplest form. After rationalizing the denominator using the conjugate method, the expression will likely be a more complex fraction.

• Look to reduce numerators and denominators.
• Check if common factors exist to simplify further.
• Simplify any radicals if possible.

Finally, combining the simplified numerator and new denominator gives the fully simplified fraction. In the case of the original problem, after rationalizing, your fraction becomes \(\frac{8(1-\sqrt{10})}{-9}\), which simplifies finally into \(-\frac{8}{9} + \frac{8\sqrt{10}}{9}\). This process aids in making readability and subsequent calculations much easier, ensuring each step is clear and logical.