Problem 63 - Algebra 1

Question

$$\text { Simplify } \sqrt{5}(6+\sqrt{5})^{2}$$ $$\begin{array}{llllll} \text { (A) } 41+2 \sqrt{5} & \text { (B) } 53 \sqrt{5} & \text { (C) } 41 \sqrt{5}+60 & \text { (D } 101 \sqrt{5} \end{array}$$

Step-by-Step Solution

Answer

The correct answer is 101 \(\sqrt{5}\). This means option D is the most accurate solution to the problem

Step 1: Expand the Term

Begin by expanding the term \((6+ \sqrt{5})^2 \). You can do this by applying the formula \(a^2 + 2ab + b^2\) where \(a\) is 6 and \(b\) is \(\sqrt{5}\).

Step 2: Multiply each term

Next, multiply each term by \(\sqrt{5}\). Obtain \(\sqrt{5} \times 36\), \(\sqrt{5} \times 12\sqrt{5}\) and \(\sqrt{5} \times 5\). While carrying out these operations remember that \(\sqrt{5}\) squared gives \(5\).

Step 3: Simplify the expression

Combine all three terms together and simplify the expression yielding the final answer.

Key Concepts

Radicals

Radicals or roots are essential tools in algebra that allow us to express numbers in a different form. Typically, a radical is represented by a square root symbol, but it can also denote cube roots or higher orders.

The most prevalent type used is the square root, seen as the operation extracting the original number. Whenever we have a square root, such as \(\sqrt{5}\), it implies that we look for a number which, when squared, gives 5. Learning how to work with radicals involves understanding their arithmetic properties. For instance, when multiplying radicals, you can utilize the property \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\).

Another key aspect is simplifying radicals whenever possible. This step can sometimes simplify calculations or expressions in exercises. Realize that \(\sqrt{5}\) is already in its simplest form as 5 is prime. Remember, though, that squaring a radical returns it to the number under the root, such as \(\sqrt{5} \times \sqrt{5} = 5\).

• Use radicals wisely as they can both simplify and complicate solutions depending on how they are handled in equations.

Polynomial Expansion

Expanding a polynomial is a foundational skill in algebra. When given a binomial raised to a power like \((6 + \sqrt{5})^2\), the goal is to write it as a simplified polynomial.

The binomial theorem is a handy tool here. For the expression \((a + b)^2\), we employ the formula \(a^2 + 2ab + b^2\). This formula helps us breakdown expressions into manageable parts. For the given exercise, substituting \(a = 6\) and \(b = \sqrt{5}\) gives:

• \(a^2 = 36\)
• \(2ab = 12\sqrt{5}\)
• \(b^2 = 5\)

Adding these components, we achieve a fully expanded polynomial.

Understanding polynomial expansion:

• Facilitates working with equations in a manageable manner.
• Turns complex expressions into a simpler, expanded form.
• Allows us to identify individual components easily.

Trigonometry

While the given exercise doesn't explicitly involve trigonometric functions, understanding how trigonometry relates to algebra is crucial.

Trigonometry often involves solving equations with variables, such as in polynomials, particularly when functions like \(\sin(x)\) or \(\cos(x)\) are expanded in terms of radicals.

A fundamental skill in trigonometry is learning how angles and sides are interrelated in a triangle through these functions. Working with radicals is commonplace here, as angles in radians and trigonometric identities frequently use these concepts. For example, \(\sin^2(x) + \cos^2(x) = 1\) can be seen analogous to working with radicals squared: \(\sqrt{x}^2 = x\).

Trigonometric knowledge empowers students to solve various problems where algebra interfaces with angles and triangles, blending verifies calculations with trigonometric identities that sometimes call for radical manipulation.