Problem 16 - Calculus Early Transcendentals: Pearson New International Edition

Question

Simplify as much as possible. Be sure to remove all parentheses and reduce all fractions. \((\sqrt{5}-\sqrt{3})^{2}\)

Step-by-Step Solution

Answer

The expression simplifies to \(8 - 2\sqrt{15}\).

Step 1: Recognize the Expression Type

The given expression \((\sqrt{5}-\sqrt{3})^{2}\) is a binomial (two terms) raised to a power. This is an example of a squared binomial, which can be expanded using the formula for the square of a binomial.

Step 2: Apply the Binomial Square Formula

The square of a binomial \((a - b)^2\) is given by the formula: \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = \sqrt{5}\) and \(b = \sqrt{3}\). Substitute these into the formula: \((\sqrt{5})^2 - 2 \cdot \sqrt{5} \cdot \sqrt{3} + (\sqrt{3})^2\).

Step 3: Simplify Each Term

Calculate each term separately: - \((\sqrt{5})^2 = 5\).- \(-2 \cdot \sqrt{5} \cdot \sqrt{3} = -2 \cdot \sqrt{15}\).- \((\sqrt{3})^2 = 3\).

Step 4: Combine Like Terms

Combine the results from step 3. The expression becomes: \[5 - 2\sqrt{15} + 3\]. Add the constants: \(5 + 3 = 8\), so the simplified expression is \[8 - 2\sqrt{15}\].

Key Concepts

Square of a Binomial

When you come across a binomial expression like \(( a - b)^2\), it may seem complex, but it's actually a straightforward formula you can use to expand it. This is known as the square of a binomial. This formula helps you break down the expression into simpler parts that are easy to manage.
The general formula looks like this:

• \((a - b)^2 = a^2 - 2ab + b^2\)

To apply this to our example, \((\sqrt{5} - \sqrt{3})^{2}\), we identify \(a = \sqrt{5}\) and \(b = \sqrt{3}\). Plugging these into our formula, we get:

• \((\sqrt{5})^2 - 2 \cdot \sqrt{5} \cdot \sqrt{3} + (\sqrt{3})^2\)

By using this formula, you're simply expanding the expression so it's easier to deal with each term, setting the stage for simplification.

Simplification

Once you've expanded the square of a binomial, the next goal is to simplify the expression. This means reducing it as much as possible while making it easier to understand or work with. To do this, you need to handle each term separately and combine them in clear steps.
In the expression \((\sqrt{5})^2 - 2 \sqrt{5} \cdot \sqrt{3} + (\sqrt{3})^2\), simplification involves:

• Calculating \((\sqrt{5})^2 = 5\)
• Calculating \(-2 \cdot \sqrt{5} \cdot \sqrt{3} = -2 \sqrt{15}\)
• Calculating \((\sqrt{3})^2 = 3\)

After obtaining these results, combining the constant terms (5 and 3) gives us 8, which simplifies the binomial to \[8 - 2\sqrt{15}\]. This step-by-step approach ensures clarity and precision.

Radical Expressions

Radical expressions, like \(\sqrt{5}\) or \(\sqrt{3}\), are used frequently in binomials. They can seem tricky at first, but understanding how to manipulate them is key.
A radical expression is essentially any expression that involves square roots, cube roots, or other roots. In our binomial expression, managing these requires combining basic radical rules with arithmetic:

• The square of a square root, like \((\sqrt{5})^2\), simplifies directly to 5 because the square and the square root cancel each other out.
• When multiplying two distinct square roots, such as \(\sqrt{5} \cdot \sqrt{3}\), the product is the square root of the product: \(\sqrt{15}\).

By following these principles, you can handle radical expressions smoothly. Mastery of these concepts allows you to simplify and transform binomial expansions and other algebraic tasks easily.