Problem 178 - Elementary Algebra

Question

In the following exercises, simplify. $$ \sqrt{48}+2 \sqrt{3} $$

Step-by-Step Solution

Answer

\[\root{48} + 2 \root{3} = 6 \root{3}\]

Step 1 - Simplify the Square Root

First, simplify \(\root{48}\). Find the prime factorization of 48 to identify perfect squares. \(48 = 2 \times 24 = 2 \times 2 \times 12 = 2 \times 2 \times 2 \times 6 = 2 \times 2 \times 2 \times 2 \times 3 = 2^4 \times 3\). Therefore, \(\root{48} = \root{2^4 \times 3} = \root{2^4} \times \root{3} = 2^2 \times \root{3} = 4 \root{3}\).

Step 2 - Combine Like Terms

Now combine like terms: \[ \root{48} + 2 \root{3} = 4 \root{3} + 2 \root{3}. \] Since both are like terms, simply add the coefficients: \[ 4 \root{3} + 2 \root{3} = (4 + 2) \root{3} = 6 \root{3}. \]

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a composite number into its prime factors. A prime number is one that only has two distinct positive divisors: 1 and itself. To simplify square roots, it's essential to find the prime factors of the number under the square root.

• For example, to factorize 48, we start by dividing by the smallest prime number, 2:
• 48 ÷ 2 = 24
• 24 ÷ 2 = 12
• 12 ÷ 2 = 6
• 6 ÷ 2 = 3

Now, we can write 48 as the product of prime factors:

• 48 = 2 × 2 × 2 × 2 × 3 = 2^4 × 3

Understanding prime factorization helps simplify square roots by identifying perfect squares within the number. It's a fundamental tool in number theory and helps in a variety of mathematical problems.

Perfect Squares

Perfect squares are numbers that can be expressed as the product of an integer with itself. For example, 1, 4, 9, 16, 25, and 36 are perfect squares since:

• 1 = 1 × 1
• 4 = 2 × 2
• 9 = 3 × 3
• 16 = 4 × 4
• 25 = 5 × 5
• 36 = 6 × 6

When simplifying square roots, it helps to find and separate any perfect squares within the number. Taking the square root of a perfect square eliminates the radical:

• For instance, \( \root{48} = \root{2^4 \times 3} \). Since \( 2^4 \) is a perfect square (\( 2^4 = (2^2)^2 \)), we can simplify it to 4.
• Therefore, \( \root{48} = \root{2^4 \times 3} = 4\root{3} \).

Recognizing perfect squares makes it easier to simplify expressions involving square roots and reduces them to simpler forms.

Combining Like Terms

Combining like terms is an essential step in simplifying algebraic expressions. In the context of square roots, like terms are those that have the same radical part. For example:

• In the expression \( \root{48} + 2 \root{3} \), we simplify \( \root{48} \) to \( 4 \root{3} \), giving us \( 4 \root{3} + 2 \root{3} \).
• Since the terms \( 4 \root{3} \) and \( 2 \root{3} \) have the same radical part (\( \root{3} \)), they are like terms and can be combined.
• Add the coefficients (numbers in front of the radicals): \( 4 + 2 = 6 \).
• So, \( 4 \root{3} + 2 \root{3} = 6 \root{3} \).

Combining like terms reduces the expression to its simplest form, making it easier to understand and work with. Pay attention to the coefficients and common radical parts when simplifying these kinds of expressions.