Problem 117 - Elementary and Intermediate Algebra

Question

Simplify each radical expression, if possible. Assume all variables are unrestricted. $$ \sqrt{n^{2}+12 n+36} $$

Step-by-Step Solution

Answer

The simplified form of the expression is \(n+6\).

Step 1: Identify the Radical Expression Type

The given expression is \( \sqrt{n^{2}+12 n+36} \). It is a square root of a polynomial expression, which can potentially be simplified.

Step 2: Check for Perfect Square Trinomial

A perfect square trinomial takes the form \((a+b)^2 = a^2 + 2ab + b^2\). Compare this form with the given expression \(n^{2} + 12n + 36\) to see if it matches the formula for a perfect square.

Step 3: Determine Components of the Perfect Square

To match \(n^{2} + 12n + 36\) with \((n+b)^2 = n^2 + 2nb + b^2\), observe that \(2nb = 12n\) gives \(b = 6\). Moreover, \(b^2 = 36\) is consistent with \(b = 6\).

Step 4: Express as a Square

The trinomial \(n^{2} + 12n + 36\) can be written as \( (n+6)^2 \).

Step 5: Simplify the Radical Expression

Since \((n+6)^2\) is a perfect square, its square root is the expression itself without the square: \(\sqrt{ (n+6)^2 } = n+6\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a type of polynomial expression that can be factored into a square of a binomial. Let’s break it down:

• It has a standard form: \((a+b)^2 = a^2 + 2ab + b^2\).
• This form indicates that the trinomial results from squaring a simple expression \((a+b)\).

Identifying a perfect square trinomial involves recognizing this pattern in any polynomial. You can do this by checking each part:

• Compare the first term with \(a^2\) and the last term with \(b^2\).
• Ensure the middle term is twice the product of the terms that would square to give \(a\) and \(b\); that is, check if the middle term matches \(2ab\).

In the original exercise, the expression \(n^2 + 12n + 36\) fits this pattern perfectly:

• \(a^2 = n^2\), so \(a = n\).
• \(b^2 = 36\), so \(b = 6\).
• Middle term is \(12n\), which is \(2nb\) confirming it as \((n+6)^2\).

This recognition of the pattern allows us to express the trinomial as \((n+6)^2\) easily.

Square Root of Polynomial

Finding the square root of a polynomial, especially a perfect square trinomial, simplifies significantly when you recognize it as a squared expression.

• The idea is to "undo" the square operation.
• When you square a number or expression, you create a perfect square.
• When taking the square root of a perfect square, you return to the original number or expression.

In our exercise:

• \(\sqrt{(n+6)^2}\) is essentially asking, "What squared equals \((n+6)^2\)?"
• The answer, of course, is \(n+6\).

Thus, finding the square root of a polynomial like this, when it is already a perfect square trinomial, becomes a straightforward simplification process. It is crucial to verify first that the polynomial is indeed a perfect square before simplification.

Radical Simplification Process

Simplifying radicals involves reducing a radical expression into its simplest form. This process is vital in mathematics as it makes expressions easier to work with. Here, the goal is to present the expression in a simpler or more usable form.

• First, identify if the expression under the radical is a perfect square.
• If it is, like in our example with \(\sqrt{n^{2}+12 n+36}\) simplifying becomes a matter of recognizing it as \(\sqrt{(n+6)^2}\).
• In simplifying this, since \((n+6)^2\) is already a perfect square, you can directly simplify the given expression to \(n+6\).

Simplifying radicals may often require factorization or recognizing patterns:

• Look for terms that can be squared to form parts of the radical expression.
• Use these observations to simplify accurately and efficiently.

By mastering this process, handling complex algebraic expressions or equations becomes much easier.