Problem 13 - Prealgebra

Question

Simplify each square root, then combine if possible. Assume all variables represent positive numbers. $$\sqrt{75}+\sqrt{27}$$

Step-by-Step Solution

Answer

The simplified expression is \( 8\sqrt{3} \).

Step 1: Simplify \( \sqrt{75} \)

First, consider the prime factorization of 75. It can be expressed as the product of its prime factors: \( 75 = 3 \, \times \, 5^2 \). Apply the square root to each factor: \( \sqrt{75} = \sqrt{3 \, \times \, 5^2} = \sqrt{3} \, \times \, \sqrt{5^2} = \sqrt{3} \, \times \, 5 = 5\sqrt{3} \).

Step 2: Simplify \( \sqrt{27} \)

Next, find the prime factorization of 27. It can be written as \( 27 = 3^3 \). Take the square root of this product: \( \sqrt{27} = \sqrt{3^3} = \sqrt{3 \, \times \, 3^2} = \sqrt{3} \, \times \, \sqrt{3^2} = 3\sqrt{3} \).

Step 3: Combine Like Terms

Now, combine \( 5\sqrt{3} \) and \( 3\sqrt{3} \) since they have the same radical part: \( 5\sqrt{3} + 3\sqrt{3} = (5 + 3)\sqrt{3} = 8\sqrt{3} \).

Key Concepts

Understanding Prime Factorization

Prime factorization is a method used to express a number as the product of its prime numbers. Prime numbers are like the building blocks of all numbers—they cannot be broken down into smaller, whole-number factors other than 1 and themselves. For instance, the number 75 can be broken down into its prime factors.

• Start with the smallest prime number, which is 2. Since 75 is not even, it isn’t divisible by 2.
• Next, try dividing by 3. Since 75 divided by 3 is 25, we know that 3 is a prime factor.
• Now take 25, and notice that it can be divided by 5. In fact, 25 is 5 squared, or 5 times 5, making 5 another prime factor, but repeating.
• This gives us the prime factorization: \( 75 = 3 \times 5^2 \).

In the example of 27, you see how it only breaks down with 3 as a factor. By dividing 27 by 3 repeatedly, you derive its prime factorization: \( 27 = 3^3 \). Recognizing these prime factors is essential in simplifying square roots.

Combining Like Terms in Radicals

Combining like terms involves adding or subtracting terms that have the exact same variable parts. In the realm of radicals, this process is similar but instead deals with the parts under the square root sign.

• The phrase "like terms" refers to terms that possess the same radical component — the number or expression under the square root.
• Look at \( 5\sqrt{3} \) and \( 3\sqrt{3} \). Since the radical part, \( \sqrt{3} \), is identical in both, they can be combined.
• This is akin to adding apples with apples: \( 5\sqrt{3} + 3\sqrt{3} = (5 + 3)\sqrt{3} = 8\sqrt{3} \).

By focusing on the similar radical parts, we streamline expressions into more manageable forms, simplifying calculations and evaluations.

Steps to Simplify Radicals

Radicals simplification refers to the process of finding the simplest or most straightforward form of a square root expression. This simplification is key when dealing with expressions involving square roots, making them easier to use in further calculations.

• Start by finding the prime factorization of the number under the square root, as explored with 75 and 27 above.
• Once the number is expressed as a product of prime factors, pair the factors when possible. Each pair of identical factors can "escape" the square root as a single number because the square root of a product of numbers is the product of the square roots of the numbers.
• For example, \( \sqrt{75} = \sqrt{3 \times 5^2} \) becomes \( 5\sqrt{3} \) as \( 5^2 \) simplifies out of the radical.
• Similarly, \( \sqrt{27} = \sqrt{3^3} = 3\sqrt{3} \) as the pair of 3s is taken out as a single number.

This systematic approach to radicals helps create simplified expressions that are quicker and less error-prone during calculations.