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Prime factorization is the process of breaking down a composite number into its fundamental building blocks, which are prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.
For example:

• 2, 3, 5, 7, and 11 are prime numbers.
• Composite numbers like 84 can be factored into prime numbers.

In our exercise, 84 was decomposed through prime factorization as shown:

1. First, identify that 84 is even and divisible by 2.
   Dividing 84 by 2 gives us 42.
2. 42 is also even, so divide it by 2 again.
   This yields 21.
3. Now, 21 is not even, but it's divisible by 3 (since 2 + 1 = 3, a multiple of 3).
   Dividing 21 by 3 gives 7.
4. Finally, 7 is itself a prime number.

Thus, the prime factorization of 84 is expressed as \(2^2 \times 3 \times 7\). This process is essential in simplifying radicals, as it allows us to easily spot pairs of the same number that can leave the square root.

The properties of square roots are fundamental in understanding how to simplify expressions involving radicals. One key property is that the square root of a product is equal to the product of the square roots of each factor. This can be written as:
\[\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\]This property is shown in our problem with the expression \(\sqrt{84}\).
Using prime factorization, we rewrite 84 as \(2^2 \times 3 \times 7\) and apply the square root property:

• \(\sqrt{2^2 \times 3 \times 7} = \sqrt{2^2} \times \sqrt{3} \times \sqrt{7}\)
• Because \(\sqrt{2^2} = 2\), the expression becomes: \(2 \times \sqrt{3} \times \sqrt{7}\)

The power of square roots allows us to simplify \(\sqrt{84}\) into \(2 \sqrt{21}\). This understanding is crucial because it helps transform more complex expressions into easier forms.

An expression in mathematics is a combination of numbers, operations, and sometimes variables. It's important to know how to manipulate these expressions to simplify them or solve them.
A key aspect of this is understanding how to apply rules and properties such as the one used in our exercise.
Take the expression \(2 \sqrt{84}\). Here, the coefficient 2 is multiplied by the square root portion which is \(\sqrt{84}\).

• The simplification starts by using prime factorization to break down 84.
• Then, properties of square roots are applied to factor and simplify \(\sqrt{84}\) into \(2 \sqrt{21}\).
• Finally, we multiply the initial coefficient of 2 by the simplified form \(2 \sqrt{21}\).

This results in the final expression \(4 \sqrt{21}\). Understanding how expressions can be manipulated in this way is an essential skill in mathematics.

Integers include all whole numbers, whether they are positive, negative, or zero. They do not include fractions or decimals, making them a crucial part of arithmetic and algebra.
In our exercise, the objective was to simplify the expression and reach a form where the result is as close to an integer as possible.
Naturally, some radicals result in non-integer expressions, but practicing simplification can often help in approximating or expressing results with integer components.
For example, when simplifying \(\sqrt{84}\) using prime factors, we find that part of it can be approximated with integers as far as the perfect square components (like \(2^2\)) are concerned.

• In our simplification, \(\sqrt{84}\) becomes \(2 \sqrt{21}\).
• Multiply by 2 to fully simplify as \(4 \sqrt{21}\).

While \(\sqrt{21}\) remains irrational, the coefficient 4 is an integer result of this simplification process. This blend of integers and radicals is a common outcome in math problems.