Problem 49 - Intermediate Algebra

Question

Rationalize each denominator. See Example 4. $$ \frac{8}{1+\sqrt{10}} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Conjugate

To rationalize the denominator, we identify the conjugate of the denominator. The conjugate of \(1 + \sqrt{10}\) is \(1 - \sqrt{10}\).

Step 2: Multiply by the Conjugate

Multiply the numerator and the denominator by the conjugate of the denominator: \[\frac{8}{1 + \sqrt{10}} \times \frac{1 - \sqrt{10}}{1 - \sqrt{10}}\]

Step 3: Simplify the Numerator

The new numerator after multiplication becomes:\[8 \times (1 - \sqrt{10}) = 8 - 8\sqrt{10}\]

Step 4: Simplify the Denominator

The new denominator after multiplication became:\[(1 + \sqrt{10})(1 - \sqrt{10}) = 1^2 - (\sqrt{10})^2 = 1 - 10 = -9\]

Step 5: Write the Rationalized Expression

Combine the results from Step 3 and Step 4 to write the rationalized fraction:\[\frac{8 - 8\sqrt{10}}{-9}\]

Step 6: Simplify the Expression

Separate the expression into two terms:\[\frac{8}{-9} - \frac{8\sqrt{10}}{-9} = -\frac{8}{9} + \frac{8\sqrt{10}}{9}\]

Key Concepts

Conjugate Multiplication

When working to rationalize a denominator, one of the key steps involves multiplying by the conjugate. But what is a conjugate? For any binomial expression in the form of \(a + b\), the conjugate is \(a - b\), and vice versa. By multiplying the numerator and the denominator by this conjugate, we effectively work to eliminate the radical expression from the denominator altogether.

• This process relies on the identity \((a + b)(a - b) = a^2 - b^2\), which is based on the difference of squares.
• In our problem, the expression in the denominator, \(1 + \sqrt{10}\), has the conjugate \(1 - \sqrt{10}\).

By applying conjugal multiplication, you turn the complex expression into a simpler, rational form. This allows us to evaluate or further simplify without the complications of a square root in the denominator.

Simplifying Radicals

Radicals can make mathematical expressions look complicated, and they often need to be simplified for ease of calculation. Simplifying radicals involves expressing them in a form where no perfect square factors greater than 1 remain under the square root symbol.

• In the context of our problem, we look at the denominator \((1 + \sqrt{10})(1 - \sqrt{10})\), which becomes \(1 - 10\).
• This simplification relies on the difference of squares identity, resulting in a neat and clean expression: \(-9\).

This key step not only simplifies our results but also helps in rationalizing the denominator, rendering it entirely free of square roots.

Numerator and Denominator Manipulation

Rationalizing often involves a lot of algebraic manipulation of both the numerator and the denominator. The goal is to achieve an expression that is easier to understand or compute, typically with a rational number in the denominator.

• First, we multiply the numerator and the denominator by the conjugate to deal with the term under the square root. This results in transforming the fractions into simpler components.
• For instance, the numerator \(8(1 - \sqrt{10})\) becomes \(8 - 8\sqrt{10}\), as seen in our step-wise solution.
• The denominator, simplified through conjugate multiplication, results in a straightforward integer \(-9\), as shown in the final fraction: \(\frac{8 - 8\sqrt{10}}{-9}\).

Finally, we break this fraction down to its simplest form, separating into two distinct terms: \(-\frac{8}{9} + \frac{8\sqrt{10}}{9}\). By manipulating the numerator and denominator, complex expressions turn friendly and understandable.