Problem 100 - Algebra 2

Question

Simplify. Rationalize all denominators. $$ (\sqrt{8}-\sqrt{7})^{2} $$

Step-by-Step Solution

Answer

The simplified and fully rationalized version of the expression is \(15 - 4\sqrt{14}\).

Step 1: Simplify Radical

First, break down \(\sqrt{8}\) into \(\sqrt{4}* \sqrt{2} = 2\sqrt{2}\). Thus the new expression would be \( (2\sqrt{2} - \sqrt{7})^{2} \)

Step 2: Square the Expression

Next, square the entire expression. Using the formula (a - b)^{2} = a^{2} - 2ab + b^{2}, we get \(4*2 - 2*2*\sqrt{2}* \sqrt{7} + 7 = 8 - 4\sqrt{14} + 7 = 15 - 4\sqrt{14}.\)

Step 3: Rationalize if Necessary

A final review of the expression finds no denominators to rationalize. The final simplified expression is \(15 - 4\sqrt{14}\).

Key Concepts

Simplifying Radicals

Understanding radicals is important in algebra. Radicals often involve square roots. We encounter them when we want to simplify expressions like \( \sqrt{8} \).
To simplify a radical, break it down into smaller parts. Find a perfect square factor of the number under the radical (also known as the radicand). For example, since the number 8 has the perfect square factor 4, we can express its square root as \( \sqrt{8} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2} \).

The method involves these key steps:

• Identify a largest perfect square as a factor of the radicand.
• Rewrite the radical as the product of two square roots.
• Simplify the expression by pulling out of the square root any factors that are perfect squares.

This process helps to simplify the overall expression and is especially useful when making calculations more manageable.

Binomial Expansion

Binomial expansion involves expressing the square of a binomial in expanded form. This is guided by the formula \((a - b)^2 = a^2 - 2ab + b^2 \).
When faced with the expression \((2\sqrt{2} - \sqrt{7})^2\), we use this formula to expand and simplify.

Here is how the expansion works in this scenario:

• First, square the first term \((a)\) which results in \((2\sqrt{2})^2 = 8\).
• Second, calculate \(2ab\), which involves multiplying \(2\times 2\sqrt{2} \times \sqrt{7}\) to get \(-4\sqrt{14}\).
• Finally, square the second term \((b)\) resulting in \(\sqrt{7}^2 = 7\).

The expanded expression results in \(8 - 4\sqrt{14} + 7\), which further simplifies to \(15 - 4\sqrt{14}\). Through such expansion, complex expressions become straightforward by tackling each part of the binomial one by one.

Exponent Rules

Exponent rules are foundational in understanding algebraic expressions. These rules help us to simplify expressions by applying specific guidelines on exponents.
When dealing with the expression \((2\sqrt{2} - \sqrt{7})^2\), exponent rules ensure that we multiply exponents correctly while maintaining order of operations.

Key exponent rules to apply include:

• The power of a product rule, which states \((ab)^n = a^n \cdot b^n\). This means both terms inside a parenthesis are raised to the power when squared.
• The power of a power rule, which tells us \((a^m)^n = a^{m \cdot n}\). This is applied when simplifying expressions with nested exponents.

Through an understanding and practice of these rules, large and seemingly complex expressions become simple to manage. It enhances our confidence in dealing with algebraic tasks consequently, making the solving process efficient and orderly.