Problem 74 - Calculus with Concepts in Calculus

Question

a. Show that \(1 /(\sqrt{25,000}-\sqrt{24,998})=\frac{1}{2}(\sqrt{25,000}\) \(+\sqrt{24,998})\) b. Compare the value your calculator gives for \(1 /(\sqrt{25,000}-\sqrt{24,998})\) with the value it gives for \(\frac{1}{2}(\sqrt{25,000}+\sqrt{24,998})\). If they are different, which do you think is the more accurate?

Step-by-Step Solution

Answer

The expressions are equal, as shown in the solution steps. Use a calculator to confirm they give the same value.

Step 1: Understand the Problem

We want to show that the expressions \( \frac{1}{\sqrt{25,000} - \sqrt{24,998}} \) and \( \frac{1}{2}(\sqrt{25,000} + \sqrt{24,998}) \) are equal. Moreover, we'll compare their calculated values using a calculator.

Step 2: Simplify the Expression

Notice that the expression \( \frac{1}{\sqrt{25,000} - \sqrt{24,998}} \) can be simplified by multiplying the numerator and the denominator by the conjugate of the denominator: \( \sqrt{25,000} + \sqrt{24,998} \).

Step 3: Apply Conjugate Multiplication

Multiply the numerator and denominator of \( \frac{1}{\sqrt{25,000} - \sqrt{24,998}} \) by \( \sqrt{25,000} + \sqrt{24,998} \) to get:\[ \frac{( \sqrt{25,000} + \sqrt{24,998})}{ (\sqrt{25,000} - \sqrt{24,998})(\sqrt{25,000} + \sqrt{24,998})} \]

Step 4: Simplify the Denominator

The denominator becomes a difference of squares: \( (\sqrt{25,000})^2 - (\sqrt{24,998})^2 = 25,000 - 24,998 = 2 \).Thus, we have:\[ \frac{\sqrt{25,000} + \sqrt{24,998}}{2} \]

Step 5: Confirm the Match

Observe that the expression \( \frac{\sqrt{25,000} + \sqrt{24,998}}{2} \) is the same as \( \frac{1}{2}(\sqrt{25,000} + \sqrt{24,998}) \). This completes the proof for part (a) of the question.

Step 6: Calculate and Compare Values

Use a calculator to compute both \( \frac{1}{\sqrt{25,000} - \sqrt{24,998}} \) and \( \frac{1}{2}(\sqrt{25,000} + \sqrt{24,998}) \). Verify whether the numeric results are equal.

Key Concepts

Conjugate Multiplication

Conjugate multiplication is a useful algebraic tool that helps us simplify expressions involving square roots.
The idea behind this technique is to eliminate radicals in the denominator by multiplying by the conjugate.

Here's how it works:

• Identify the conjugate of the denominator, which is obtained by changing the sign between two terms. For instance, the conjugate of \( \sqrt{25,000} - \sqrt{24,998} \) is \( \sqrt{25,000} + \sqrt{24,998} \).
• Multiply both the numerator and the denominator by this conjugate. The process leverages the property that multiplying by the conjugate forms a difference of squares, simplifying the expression.

By using the conjugate, we are able to transform a complex expression into a more manageable form, making it easier to handle in calculations.
The example in the exercise shows that this technique can simplify a fraction such that the radicals cancel out in the denominator.

Difference of Squares

The difference of squares is a fundamental algebraic identity used frequently in mathematics.
It is written as:

• \( a^2 - b^2 = (a-b)(a+b) \)

This identity is particularly useful when dealing with expressions involving square roots. If you recognize a difference of squares, you can simplify expressions in one swift step.

In the context of the original exercise, when we multiply \( \sqrt{25,000} - \sqrt{24,998} \) by its conjugate, we effectively apply the difference of squares formula:

• \( (\sqrt{25,000} - \sqrt{24,998})(\sqrt{25,000} + \sqrt{24,998}) = (\sqrt{25,000})^2 - (\sqrt{24,998})^2 = 25,000 - 24,998 \)

By applying this identity, we simplified the expression greatly, arriving at the solution quickly. This method leverages the symmetry and simplicity within algebraic structures. Importantly, it replaces potentially complex calculations with straightforward arithmetic, increasing efficiency.

Calculation Accuracy

Calculation accuracy is critical in mathematics, especially when dealing with intricate expressions.
Even small errors can lead to large discrepancies in results, emphasizing the importance of using precise methods.

In this exercise, we demonstrated that simplifying \( \frac{1}{\sqrt{25,000} - \sqrt{24,998}} \) using algebraic techniques can lead to more accurate calculations.

Here are some key points:

• When you directly compute \( \frac{1}{\sqrt{25,000} - \sqrt{24,998}} \) on a calculator, it might yield a different result due to limited precision or rounding errors inherent in numerical computation.
• In contrast, the simplified version \( \frac{1}{2}(\sqrt{25,000} + \sqrt{24,998}) \) reveals the underlying symmetry and is free from such numerical instabilities.

Thus, applying algebraic methods not only helps arrive at an exact expression but also protects against potential inaccuracies during calculations.
This validation of precision by simplification illustrates the power of algebra in ensuring accurate mathematical results.