Problem 18 - Algebra 2

Question

Multiply. $$ (2 \sqrt{5}+3 \sqrt{2})^{2} $$

Step-by-Step Solution

Answer

The simplified form of \((2\sqrt{5} + 3\sqrt{2})^2\) is \(38 + 12\sqrt{10}\).

Step 1: Identify the terms in the binomial

Here, \( a = 2\sqrt{5} \) and \( b = 3\sqrt{2} \). The task is to square this binomial.

Step 2: Apply the binomial square identity

According to the identity, (a+b)^2= a^2 + 2ab + b^2. Substituting the given values, we get \((2\sqrt{5} + 3\sqrt{2})^2 = (2\sqrt{5})^2 + 2 * 2\sqrt{5} * 3\sqrt{2} + (3\sqrt{2})^2 \).

Step 3: Simplify each term

Simplify each term in the equation separately. We get 20 + 12\sqrt{10} + 18 = 38 + 12\sqrt{10} .

Key Concepts

Square of Binomials

When you come across the square of a binomial, think of it as a simple pattern you can use to simplify expressions. A binomial is an expression that has two terms, like

• \((a + b)^2\)
• where "a" and "b" are any numbers or terms.

To expand the square of a binomial, you can use the formula

• \((a+b)^2 = a^2 + 2ab + b^2\).

Let’s dissect this formula:

• You first square each term in the binomial: \(a^2\) and \(b^2\).
• Then, you multiply the terms together and double it: \(2ab\).

In the problem given, your task was to find the square of \((2\sqrt{5} + 3\sqrt{2})\). The formula allows you to expand this easily by following the pattern:

• First: \((2\sqrt{5})^2 = 20\)
• Middle: \(2 \times 2\sqrt{5} \times 3\sqrt{2} = 12\sqrt{10}\)
• Last: \((3\sqrt{2})^2 = 18\)

Putting it all together, the expanded form is \(20 + 12\sqrt{10} + 18 = 38 + 12\sqrt{10}\). It’s really just following the formula, step by step.

Radicals

Radicals can seem tricky, but they are just special symbols for roots, like square roots. The radical sign, \(\sqrt{}\), indicates the square root of a number. When multiplying or squaring terms involving radicals, you should follow these key rules:

• Squaring a square root returns the original number. For example, \((\sqrt{5})^2 = 5\).
• When multiplying square roots, multiply the numbers inside the radicals. For example, \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\).

In our exercise, working with terms like \(2\sqrt{5}\) and \(3\sqrt{2}\) simply requires neat multiplication. Consider multiplying \(2\sqrt{5}\) by itself:

• This becomes \(4 \times 5 = 20\) using the rule \((\sqrt{a})^2 = a\).

Similarly, \(3\sqrt{2} \times 3\sqrt{2}\) is computed as

• \(9 \times 2 = 18\).

For the middle term, \(2 \cdot 2\sqrt{5} \cdot 3\sqrt{2}\), follow these steps:

• Multiply the coefficients: \(2 \times 2 \times 3 = 12\).
• Multiply the radicals: \(\sqrt{5} \times \sqrt{2} = \sqrt{10}\).

Thus, this term equals \(12\sqrt{10}\), bringing together the multiplication rules for radicals.

Simplification of Expressions

Simplification involves combining like terms to create the simplest form of an expression. In the case of our problem, simplifying an expression with radicals and integers comes next. Let's discuss what we need to watch out for:

• Make sure to simplify each term completely. For instance, \(20\) and \(18\) are straightforward; they do not require further simplification.
• Look at the radicals. Here, our combinations result in non-simplifiable radicals like \(12\sqrt{10}\).
• Add up the integers. Combine all whole number terms: \(20 + 18 = 38\).

So, after perfect simplification here, the final expression becomes:

• \(38 + 12\sqrt{10}\).

Resist the temptation to further break radicals unless they can combine to form a simple radical. In this solution, no further work can be done beyond recognizing that the expression is as simple as possible. By paying attention to these principles, you ensure clarity and correctness every time you handle similar problems.