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 simplified expression is \(8 - 2\sqrt{15}\).

Step 1: Apply Binomial Theorem

The expression \((\sqrt{5}-\sqrt{3})^{2}\) is a perfect square. According to the binomial theorem, it can be expanded as \((a-b)^{2} = a^{2} - 2ab + b^{2}\). Here, \(a = \sqrt{5}\) and \(b = \sqrt{3}\).

Step 2: Square Each Term

Calculate \(a^{2}\) and \(b^{2}\):\(a^{2} = (\sqrt{5})^{2} = 5\)\(b^{2} = (\sqrt{3})^{2} = 3\)

Step 3: Calculate the Product of Two Terms

Find \(2ab\):\(2ab = 2 \times \sqrt{5} \times \sqrt{3} = 2 \times \sqrt{15}\)

Step 4: Substitute and Simplify

Substitute the values from the previous steps into the binomial expansion formula:\((\sqrt{5} - \sqrt{3})^{2} = 5 - 2\sqrt{15} + 3\)Combine like terms:\(= 8 - 2\sqrt{15}\)

Key Concepts

Perfect Square

When we hear the term 'perfect square,' it refers to a number or an expression resulting from squaring another whole number or expression. The binomial theorem often helps us understand perfect squares.
For example, if we have \(x+1\) and square it, it becomes \(x^{2} + 2x + 1\), which is a perfect square polynomial.
In the given problem, \((\sqrt{5}-\sqrt{3})^{2}\) is a perfect square because it's the square of \(\sqrt{5}-\sqrt{3}\).
To simplify this, we use the formula for the square of a binomial: \(a^{2} - 2ab + b^{2}\). This is foundational when working with expressions involving square roots and other algebraic terms.

Simplification

Simplification is an essential algebraic process used to make expressions as straightforward as possible. The strategy includes combining like terms, reducing fractions, and eliminating unnecessary components.
In our specific problem \(\sqrt{5} - \sqrt{3})^{2}\), we simplified using the details from applying the binomial theorem.
We found that it reduces to \(8 - 2\sqrt{15}\).
This simplification removes all parentheses and gives us a more concise version of the expression, essential in efficient problem-solving.
Remember, each step should bring the expression closer to a basic form, which aids in understanding and calculating further mathematical operations.

Square Root

Square roots appear frequently in algebra, representing a number that, when multiplied by itself, gives a specific product.
In algebraic equations, square roots often require careful handling, especially during operations like multiplication.
For example, \(\sqrt{5}\) is the square root of 5, meaning \(\sqrt{5} \times \sqrt{5} = 5\).
During simplification, the square root terms play a crucial role, as seen in finding \(a^{2}\) and \(b^{2}\) in our problem where \(a = \sqrt{5}\) and \(b = \sqrt{3}\).
Handling square roots carefully allows you to express more complex algebraic expressions in their simplest form.

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and operations. They form the foundation of algebra and represent real-world scenarios more abstractly.
Expressions such as \(\sqrt{5}-\sqrt{3}\) incorporate constants, variables (imagined here as the radicals), and operations like subtraction and exponentiation.
When dealing with these expressions, we often look to expand or simplify as shown in the binomial expansion of \(\sqrt{5} - \sqrt{3}\).
By expanding and then simplifying expressions, we can solve equations and understand relationships between different quantities, a critical skill in navigating various algebraic challenges.