Problem 183 - Elementary Algebra

Question

In the following exercises, simplify. $$ \sqrt{48}+\sqrt{27} $$

Step-by-Step Solution

Answer

7\sqrt{3}

Step 1: Prime Factorization

Perform the prime factorization of the numbers under each square root. 48 can be factored into prime numbers as follows: 48 = 2^4 * 3. Similarly, 27 can be factored into prime numbers as follows: 27 = 3^3.

Step 2: Rewrite Square Roots Using Factorization

Rewrite each square root in terms of its prime factors. \( \sqrt{48} = \sqrt{2^4 \times 3} \) and \( \sqrt{27} = \sqrt{3^3} \)

Step 3: Simplify Each Square Root

Simplify each square root by taking out the square of each prime factor when possible. \( \sqrt{2^4 \times 3} = \sqrt{(2^2)^2 \times 3} = 2^2\sqrt{3} = 4\sqrt{3} \). Similarly, \( \sqrt{3^3} = \sqrt{3^2 \times 3} = 3\sqrt{3} \)

Step 4: Combine Like Terms

Since both of the simplified terms are multiples of \( \sqrt{3} \), combine them: \4\sqrt{3} + 3\sqrt{3} = 7\sqrt{3}.

Key Concepts

prime factorization

Prime factorization is breaking down a number into its basic building blocks. These building blocks are prime numbers, which are numbers greater than 1 that have only two factors: 1 and themselves. For example, the prime numbers 2, 3, 5, and 7 are like the Lego bricks of the number world. When you break down 48 into these prime numbers, you get 48 = 2^4 * 3. Similarly, breaking down 27 gives us 27 = 3^3.

Why do we do this? Prime factorization helps us simplify the numbers under the square root, making them easier to work with in later steps. Remember:

• Avoid skipping steps and always write down every factor.
• Check and double-check your factors to ensure accuracy.

combining like terms

Combining like terms simplifies expressions by adding or subtracting coefficients of the same variable. In this case, the variable-like part is the square root. After breaking down and simplifying the prime factors, we get two terms: 4√3 and 3√3. Both are multiples of √3, making them 'like terms.'

Why combine them? Combining like terms helps reduce the complexity of the expression, leading to fewer numbers to keep track of. Here, add the coefficients: 4 + 3, keeping √3 intact, resulting in 7√3.

• Always ensure you have simplified each term before combining.
• Adding or subtracting like terms follows standard addition or subtraction rules.

radical expressions

Radical expressions include square roots, cube roots, or any nth root. Simplifying them often involves prime factorization and combining like terms. In the exercise, we start with \(\root(48)\) and \(\root(27)\) and through prime factorization and simplification, end with 7√3.

Key points to remember:

• Always look for factors that are perfect squares (like (2^2)(2^2) for 4 or 3^2 for 9) to simplify square roots.
• Combine simplified terms when possible to arrive at a more reduced form of the expression.

Simplifying radicals makes complex problems much more manageable, turning overwhelming calculations into straightforward operations.