Problem 14 - Prealgebra

Question

Simplify each square root, then combine if possible. Assume all variables represent positive numbers. $$\sqrt{50}+\sqrt{8}$$

Step-by-Step Solution

Answer

The expression simplifies to \(7\sqrt{2}\).

Step 1: Express under Radicals with Prime Factors

First, let's express the numbers under the square roots as products of their prime factors. For 50, the factors are \(50 = 2 \times 5 \times 5\). For 8, the factors are \(8 = 2 \times 2 \times 2\).

Step 2: Simplify the Square Roots

Use the property \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\) to simplify. For \(\sqrt{50}\), use \(\sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}\). For \(\sqrt{8}\), use \(\sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}\).

Step 3: Combine the Like Terms

Now, add the simplified like terms: \(5\sqrt{2} + 2\sqrt{2}\). As both terms have \(\sqrt{2}\) in common, combine them: \((5 + 2)\sqrt{2} = 7\sqrt{2}\).

Key Concepts

Understanding Prime Factorization

Prime factorization is a technique used to express a number as the product of its prime numbers. Prime numbers are those greater than 1 that have no divisors other than 1 and themselves. To find the prime factors of a number, start by dividing it by the smallest prime number (2, 3, 5, etc.) until you reach a prime number.
\[\textbf{For example:}\]

• For 50, divide by 2 to get 25, then divide by 5 to get 5 (so, 50 = 2 x 5 x 5).
• For 8, divide by 2 repeatedly until you reach 1 (so, 8 = 2 x 2 x 2).

Using prime factorization simplifies the process of finding square roots, as it allows you to pair identical factors. Credit here goes to the prime factor's ability to show hidden pairs of numbers that can be square-rooted easily.

Exploring the Properties of Square Roots

Understanding the properties of square roots can make simplifying them easier. A crucial property is that the square root of a product is the product of the square roots: \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). This allows you to split a square root into manageable parts.
\[\textbf{Let's see it in action:}\]

• For \(\sqrt{50}\), rewrite it as \(\sqrt{25 \times 2}\). This allows you to simplify it to \(\sqrt{25} \times \sqrt{2}\), or \(5\sqrt{2}\) since \(\sqrt{25} = 5\).
• Applying the same logic to \(\sqrt{8}\), which is \(\sqrt{4 \times 2}\), gives \(\sqrt{4} \times \sqrt{2} = 2\sqrt{2}\), as \(\sqrt{4} = 2\).

Using these properties streamlines the process and aids in dealing with larger numbers, and helps in recognizing simpler forms.

Combining Like Terms in Algebra

An important skill in algebra is combining like terms, which often comes into play when simplifying expressions. Like terms are terms that contain the same variable raised to the same power. Here, when we talk about like terms, we refer to terms involving the same square root.

• Suppose you have \(5\sqrt{2}\) and \(2\sqrt{2}\). Both terms have \(\sqrt{2}\) in common, making them like terms.
• Combine them by adding the numeric coefficients: \(5\sqrt{2} + 2\sqrt{2} = (5+2)\sqrt{2} = 7\sqrt{2}\).

Combining like terms simplifies expressions and reduces their complexity. It's vital for understanding higher algebra concepts and helps in reaching the final solution more efficiently.