Problem 49 - Algebra A Combined Function

Question

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (\sqrt{5}-\sqrt{2})^{2} $$

Step-by-Step Solution

Answer

The expression simplifies to \(7 - 2\sqrt{10}\).

Step 1: Understand the Expression

The expression we need to work with is \((\sqrt{5} - \sqrt{2})^2\). This means we have to multiply \((\sqrt{5} - \sqrt{2})\) by itself.

Step 2: Apply the Binomial Theorem

Using the formula \((a - b)^2 = a^2 - 2ab + b^2\), where \(a = \sqrt{5}\) and \(b = \sqrt{2}\), we expand the expression.

Step 3: Compute Each Term Individually

First, find \(a^2 = (\sqrt{5})^2 = 5\). Second, find \(2ab = 2 * \sqrt{5} * \sqrt{2} = 2 \sqrt{10}\). Lastly, find \(b^2 = (\sqrt{2})^2 = 2\).

Step 4: Substitute and Simplify

Combine all terms to get the simplified expression: \((\sqrt{5} - \sqrt{2})^2 = 5 - 2\sqrt{10} + 2 = 7 - 2\sqrt{10}\).

Key Concepts

Simplifying Radical Expressions

Simplifying radical expressions involves breaking down expressions with roots, such as square roots, to their simplest form.
This process often includes eliminating any radicals from the denominator and expressing the simplest version of any coefficients or terms.

When simplifying, consider the following tips:

• Identify and simplify radicals by factoring out any perfect squares. For example, simplify \(\sqrt{50}\) to \(\sqrt{25 \times 2} \), which becomes \(5\sqrt{2}\).
• Combine like radicals the same way you would combine like terms in algebraic expressions. Radicals with the same index and radicand can be added or subtracted.
• Use properties of exponents if needed, such as \((a^m)^n = a^{mn}\), to simplify radicals further.

Simplification makes radical expressions easier to work with and understand in further calculations or equations.

Binomial Theorem

The Binomial Theorem is a fundamental tool in algebra that describes the expansion of powers of a binomial.
A binomial is a simple algebraic expression containing two terms. The theorem allows us to expand expressions like \((a + b)^n\) into a sum involving terms of the form \(a^{n-k}b^k\).

Specifically for our expression

• We used the case for power 2: \((a - b)^2 = a^2 - 2ab + b^2\).
• Each term is derived by selecting powers of \(a = \sqrt{5}\) and \(b = \sqrt{2}\), followed by multiplication coefficients.

The Binomial Theorem simplifies complex calculations and helps manage computations involving powers, which may otherwise be tedious.

Multiplication of Radicals

When multiplying radicals, you should be familiar with certain properties that make the process easier.
Radicals obey the rule \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\), which allows us to multiply the radicands (numbers inside the root) directly under a single radical sign.

Here are some steps to follow while multiplying radicals:

• Always check if each term can be simplified before multiplying, as this can save time and effort.
• Multiply the coefficients (numbers outside the radical) together first, if there are any.
• Then, multiply the radicands together under a single radical.

In our example, when computing \(2 \times \sqrt{5} \times \sqrt{2}\), it becomes \(2 \times \sqrt{10}\) following these principles. Ensure that the result is in its simplest form for easy interpretation.