Problem 53 - Elementary Algebra

Question

Simplify each expression by performing the indicated operation. $$ (3+\sqrt{5})^{2} $$

Step-by-Step Solution

Answer

Answer: The simplified expression is \(14 + 6\sqrt{5}\).

Step 1: Identify A and B

In this exercise, our expression is in the form \((A+B)^2\). We can see that the two terms A and B are: $$ A = 3 \\ B = \sqrt{5} $$

Step 2: Apply the binomial square formula

Now we apply the formula for squaring a binomial \((A+B)^2 = A^2 + 2AB + B^2\) with our values of A and B: $$ (3+\sqrt{5})^{2} = (3)^2 + 2(3)(\sqrt{5}) + (\sqrt{5})^2 $$

Step 3: Evaluate the terms

Next, we evaluate the squared terms and the product term individually: $$ (3)^2 = 9 \\ 2(3)(\sqrt{5}) = 6\sqrt{5} \\ (\sqrt{5})^2 = 5 $$

Step 4: Substitute and simplify

Finally, we substitute the evaluated terms back into the equation and simplify the expression: $$ (3+\sqrt{5})^{2} = 9 + 6\sqrt{5} + 5 = (9+5) + 6\sqrt{5} = 14 + 6\sqrt{5} $$ The simplified expression is: $$ 14 + 6\sqrt{5} $$

Key Concepts

Binomial Square Formula

Understanding the binomial square formula is key to simplifying expressions like \( (3+\sqrt{5})^{2} \). This formula, which is derived from the rules of exponentiation and distributive property in algebra, states that when you square a binomial (a two-term algebraic expression), the result is the square of the first term, plus two times the product of both terms, plus the square of the second term. We write it as \((A+B)^2 = A^2 + 2AB + B^2\).

When dealing with problems like the one above, we first identify the A and B terms, which are 3 and \(\sqrt{5}\) respectively. Applying the formula helps unravel the combined squared terms into individual pieces that can be handled more easily. For example, \((3)^2\) is straightforward and equals 9. The middle term, \((2AB)\), involves a multiplication of both the numerical and radical parts, resulting in \((6\sqrt{5})\). Finally, \((B^2)\) simplifies the square of a square root, which neatly dispenses with the radical, yielding 5. By carrying out these steps, we can simplify an initially complex-looking expression into a more manageable form.

Radical Expressions

Radical expressions contain roots, such as square roots or cube roots. The key to working with radical expressions is to remember that radicals represent the inverse operation of exponents. The square root of a number x, written as \(\sqrt{x}\), asks for the number which, when squared, gives x.

The square root symbol can make expressions look intimidating, but with practice, they become less daunting. In \( (3+\sqrt{5})^{2} \), the \(\sqrt{5}\) is a radical expression representing the positive number that when squared, equals 5. In the simplification process, whenever a radical is squared, as in \( (\sqrt{5})^2 \), it cancels out the square root, leaving us with the inner number, 5 in this case. Remember to maintain the distinction between a number inside the radical, like \(\sqrt{5}\), and a standalone number like 3, as each interacts differently with algebraic operations.

Algebraic Operations

Performing algebraic operations correctly is the final step to simplifying algebraic expressions. Operations such as addition, subtraction, multiplication, and exponentiation often need to be performed in a specific order, following the rules of arithmetic and algebra. This concept is essential when applying formulas like the binomial square.

In our example, after applying the binomial square formula, we perform the algebraic operations step by step. We first square the individual terms, multiply the coefficients and radical part for the middle term, and then add all the results together. In a case where numerical and radical terms are present, like \(14 + 6\sqrt{5}\), we notice that we cannot combine these terms further because they are not 'like terms'. Adding or simplifying radical terms with numerical terms would be incorrect, as they represent different mathematical quantities. Therefore, it is imperative to distinguish between these types of terms and treat them according to the rules that apply to them.