Problem 72 - College Algebra and Calculus: An Applied Approach

Question

Simplify the expression.\(2 \sqrt{108}+\sqrt{147}\)

Step-by-Step Solution

Answer

The simplified expression is \(19\sqrt{3}\).

Step 1: Simplify the first radical

2 \sqrt{108} can be simplified by breaking down 108 into its prime factors. The prime factorization of 108 is \(2 \times 2 \times 3 \times 3 \times 3 = 2^2 \times 3^3\). The square root of \(2^2\) can be taken out of the radical sign, leading to \(4\sqrt{3^3}\). Then, the square root of \(3^2\) can also be taken outside, leading finally to \(4 \times 3\sqrt{3} = 12\sqrt{3}\).

Step 2: Simplify the second radical

\(\sqrt{147}\) is simplified by breaking down 147 into its prime factors. The prime factorization of 147 is \(3 \times 7^2\). Because \(7^2\) is a perfect square, the square root of \(7^2\) can be taken out of the radical sign, leading to \(7\sqrt{3}\).

Step 3: Combine like terms

Now, you can combine \(12\sqrt{3}\) and \(7\sqrt{3}\) because they are like terms (they both contain the term \(\sqrt{3}\)). This gives the final simplified expression of \(19\sqrt{3}\).

Key Concepts

Prime Factorization

To simplify radical expressions, it's essential to understand prime factorization. This process involves breaking down a number into its basic building blocks, which are prime numbers. A prime number is a number that can only be divided by 1 and itself without leaving a remainder.
For example, when we look at 108, our task is to express it as a product of prime numbers:

• Start by dividing 108 by the smallest prime number, which is 2: 108 ÷ 2 = 54
• Continue dividing by 2 until the result is no longer even: 54 ÷ 2 = 27
• Move to the next smallest prime, which is 3: 27 ÷ 3 = 9, and 9 ÷ 3 = 3

Thus, 108 becomes \(2^2 \times 3^3\). Prime factorization helps simplify radicals by finding pairs of primes that can be extracted from under the square root. For example, in simplifying \(\sqrt{108}\), we utilize these pairs.

Perfect Squares

Recognizing perfect squares is a key step in simplifying radical expressions. A perfect square is a number that is the square of a whole number, such as 4, 9, 16, etc. It allows us to simplify the term under a square root.
For instance, let's consider \(147\):

• Its prime factorization is \(3 \times 7^2\).
• The term \(7^2\) is a perfect square.

When a number inside the radical is a perfect square, we can "take it out" of the square root. Thus, \(\sqrt{7^2}\) equals 7, simplifying \(\sqrt{147}\) to \(7\sqrt{3}\).
Spotting and using perfect squares in expressions makes simplification straightforward.

Combining Like Terms

Once you've simplified each component of a radical expression, you need to combine like terms. These terms must have the same radical part to be considered "like."
As seen in the problem, we ended up simplifying the expression to \(12\sqrt{3}\) and \(7\sqrt{3}\). Since both terms contain \(\sqrt{3}\), they are like terms and can therefore be combined:

• Add the coefficients in front of the like radical terms: 12 + 7 = 19

Thus, the simplified form of \(2 \sqrt{108} + \sqrt{147}\) is \(19\sqrt{3}\).
Combining like terms simplifies the expression further and gives a cleaner, more concise result.