Problem 16 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The square of \((\sqrt{3} + \sqrt{5})\) is \(8 + 2\sqrt{15}\)

Step 1: Identifying a and b

In this problem 'a' corresponds to \(\sqrt{3}\) and 'b' corresponds to \(\sqrt{5}\).

Step 2: Apply the binomial square formula

By applying the binomial square formula \((a + b)^2 = a^2 + 2ab + b^2\), we get: \((\sqrt{3} + \sqrt{5})^2 = (\sqrt{3})^2 + 2(\sqrt{3})(\sqrt{5}) + (\sqrt{5})^2\)

Step 3: Calculate the square of the roots

Next, compute the square of the roots to get \(3 + 2(\sqrt{3})(\sqrt{5}) + 5\)

Step 4: Simplify the expression

To further simplify the expression, multiply \(\sqrt{3}\) and \(\sqrt{5}\) to get \(3 + 2(\sqrt{15}) + 5\)

Step 5: Final Simplification

Lastly, sum the constants and rearrange the terms to get the final simplified answer of \(8 + 2\sqrt{15}\)

Key Concepts

Square Roots

Square roots are fundamental in mathematics and are often encountered when simplifying algebraic expressions. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 times 3 equals 9. In notation, the square root of a number 'x' is written as \( \sqrt{x} \). In many problems, you will encounter expressions involving square roots, especially in contexts like the binomial expansion.

When dealing with expressions like \( \sqrt{3} + \sqrt{5} \), it might look complicated, but each radical part can be treated independently. In our example, \( \sqrt{3} \) and \( \sqrt{5} \) are two separate numbers under the square root sign, which do not simplify further because they are not perfect squares. Understanding this will help you simplify expressions efficiently later on.

Simplification

Simplification is a key process in solving algebraic expressions, especially those involving square roots and exponents. In our problem, simplifying the expression \((\sqrt{3} + \sqrt{5})^2\) involves using an expansion formula to break down the expression into simpler parts.

• Using the formula \((a + b)^2 = a^2 + 2ab + b^2\), we can expand complex expressions efficiently.
• This expansion makes it possible to handle the square roots separately, simplifying the whole expression step by step.

Breaking expressions down using these methods reduces complexity. By computing the square of each square root, your work becomes straightforward.

For example, the expression \((\sqrt{3})^2 + 2(\sqrt{3})(\sqrt{5}) + (\sqrt{5})^2 = 3 + 2\sqrt{15} + 5\) simplifies to \(8 + 2\sqrt{15}\) by performing straightforward arithmetic on the constants, and the radicals are left unchanged. This type of simplification process helps in not just getting to the answer but also understanding the nature of the problem.

Algebraic Expressions

Algebraic expressions involve numbers, variables, and operations such as addition, multiplication, and extraction of roots. They are the backbone of algebra and play a significant role in solving mathematical problems. Understanding how to manipulate and simplify these expressions allows for more efficient problem-solving.

In algebraic expressions like \( (\sqrt{3} + \sqrt{5})^2 \), understanding each component is crucial. Here, \( \sqrt{3} \) and \( \sqrt{5} \) are algebraic terms that we treat like variables within the expression up until they combine or simplify.

• The core of handling algebraic expressions is knowing the formulas and rules, such as the binomial expansion, which simplify the process.
• Once you know these formulas, you can turn complex expressions into simpler forms using arithmetic operations.

Thus, algebraic expressions are not just a list of numbers and letters. They represent a method of expressing relationships and quantities algebraically in mathematics.