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Square roots are a fundamental concept in algebra.
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 the original exercise, we have the square root of 27, denoted as \( \sqrt{27} \).
Square roots can often be simplified by factoring the number under the root into its prime factors.
For instance, 27 can be factored into 3 \( \times \) 3 \( \times \) 3.
> This means: \[ \sqrt{27} = \sqrt{3 \times 3 \times 3} = 3 \sqrt{3} \]

While simplifying, you often need to rewrite square roots in their simplest form, but it's crucial to handle similar terms when they involve square roots to simplify them effectively.

In algebra, 'like terms' refer to terms that have the same variables raised to the same power.
Like terms can be combined through addition or subtraction.

In the expression \( \sqrt{27} - 3 \sqrt{27} \), both terms are 'like' because they both contain the square root of 27.

To simplify the expression, you need to factor out the like term \( \sqrt{27} \), resembling this:
\[ \sqrt{27} (1 - 3) \]

Combining like terms within the parentheses makes the operation simpler;
in this case, subtract 3 from 1, resulting in -2.
Therefore, the expression reduces to:
\[ -2 \sqrt{27} \]
Remember, combining like terms is a valuable skill that helps simplify and solve algebraic expressions efficiently.

Factoring is an essential method for simplifying algebraic expressions.
It involves breaking down an expression into simpler 'factors' that, when multiplied together, give the original expression.

In our problem, we factored out \( \sqrt{27} \) as a common factor:
\[ \sqrt{27} \left( 1 - 3 \right) \]

Simplifying the parenthesis (1 - 3) results in -2, leading to:
\[ \sqrt{27} \left( -2 \right) \rightarrow -2 \sqrt{27} \]

Factoring not only makes expressions simpler but often helps reveal other properties or solutions.
It transforms a complex problem into an easier-to-handle form.
Always look for common factors in terms to simplify when possible.