Simplifying Radical Expressions

The problem asks us to simplify the expression \(\sqrt {36{x^2} - 108x + 81}\). To simplify a square root, we often look for perfect square factors inside the radical. In this case, we have a trinomial inside the square root.

Identifying a Perfect Square Trinomial

A trinomial in the form \(ax^2 + bx + c\) might be a perfect square if it matches the pattern \((px \pm q)^2 = p^2x^2 \pm 2pqx + q^2\). Let's examine the given expression: \(36x^2 - 108x + 81\).

• The first term, \(36x^2\), is a perfect square: \((6x)^2 = 36x^2\). This suggests \(p=6\).
• The last term, \(81\), is a perfect square: \(9^2 = 81\). This suggests \(q=9\).

Now, let's check if the middle term, \(-108x\), matches \(\pm 2pqx\). Using \(p=6\) and \(q=9\):

\(2pqx = 2 \times (6x) \times 9 = 108x\).

Since the middle term is \(-108x\), the expression matches the pattern \((px - q)^2 = p^2x^2 - 2pqx + q^2\).

Therefore, the trinomial \(36x^2 - 108x + 81\) can be factored as \((6x - 9)^2\).

Simplifying the Square Root

Now we can rewrite the original expression using the factored form:

\(\sqrt {36{x^2} - 108x + 81} = \sqrt{(6x - 9)^2}\)

The square root of a squared term is the absolute value of the term:

\(\sqrt{(6x - 9)^2} = |6x - 9|\)

However, the given options are linear expressions without absolute value. In problems like this where the options are provided in this form, the simplification usually implies taking the principal root under the assumption that the expression inside the square root is non-negative, or that the context allows for \(|A|=A\). Therefore, we consider the positive root which aligns with the options:

\(\sqrt{(6x - 9)^2} = 6x - 9\)

Step-by-Step Simplification

1. Identify the expression inside the square root: \(36x^2 - 108x + 81\).
2. Recognize this trinomial might be a perfect square.
3. Find the square root of the first term: \(\sqrt{36x^2} = 6x\).
4. Find the square root of the last term: \(\sqrt{81} = 9\).
5. Check the middle term: \(2 \times (6x) \times 9 = 108x\). Since the middle term in the expression is \(-108x\), the trinomial is \((6x - 9)^2\).
6. Substitute the factored form back into the square root: \(\sqrt{(6x - 9)^2}\).
7. Simplify the square root: \(\sqrt{(6x - 9)^2} = |6x - 9|\).
8. Based on the format of the options, the simplified form is presented as \(6x - 9\).

Comparing with Options

The simplified expression \(6x - 9\) matches one of the given options.

The expression \(6x - 9\) is the correct simplified form based on the options provided.

Revision Table: Simplifying Perfect Squares

Additional Information: Absolute Value and Square Roots

When simplifying \(\sqrt{A^2}\), the result is always the absolute value of \(A\), denoted as \(|A|\). This is because the square root symbol (\(\sqrt{}\)) traditionally denotes the principal (non-negative) square root. For example, \(\sqrt{(-5)^2} = \sqrt{25} = 5\), which is \(|-5|\).

In this problem, \(\sqrt{(6x-9)^2} = |6x-9|\). The expression \(|6x-9|\) equals \(6x-9\) if \(6x-9 \ge 0\) (i.e., \(x \ge 3/2\)), and it equals \(-(6x-9) = 9-6x\) if \(6x-9 < 0\) (i.e., \(x < 3/2\)).

Since the options provided are \(6x-9\) and not \(|6x-9|\) or involving the conditional statement, it is standard practice in many contexts, particularly at introductory levels or without explicit constraints on \(x\), to present the answer as \(6x-9\), assuming the context implies the positive case or simplifies notation when options are linear.

Understanding the absolute value is crucial for full mathematical accuracy, but when multiple-choice options are limited to simple linear forms, select the one that corresponds to the expression \(A\) when simplifying \(\sqrt{A^2}\).