Problem 64 - Algebra 2

Question

Simplify each expression. $$ \sqrt{75}+\sqrt{125} $$

Step-by-Step Solution

Answer

\(\sqrt{75} + \sqrt{125} = 5\sqrt{3} + 5\sqrt{5}\)

Step 1: Prime Factorization of 75 and 125

Break down the numbers 75 and 125 into their prime factors. The prime factorization of 75 is \(3 \times 5^2\) and 125 is \(5^3\).

Step 2: Simplify the Square Roots

Next, simplify the square roots: \(\sqrt{75} = \sqrt{3\times5^2} = 5\sqrt{3}\), and \(\sqrt{125} = \sqrt{5^3} = 5\sqrt{5}\). The square root of \(5^2\) is 5 and of \(5^3\) is \(5\sqrt{5}\) as one 5 can be fully square rooted and the remaining \(\sqrt{5}\) still stays under the root.

Step 3: Final Simplification

Now, substitute these simplified square roots back into the original expression, so you get \(5\sqrt{3}+5\sqrt{5}\).

Key Concepts

Prime Factorization

Prime factorization is a method for determining the prime numbers that when multiplied together give the original number. This technique helps break down complex expressions involving square roots into simpler components. For example, when addressing the numbers 75 and 125:

• 75: We can factorize it as \(3 \times 5^2\).
• 125: This number breaks down into \(5^3\).

Understanding the prime factors is essential as it helps you recognize perfect squares within these numbers. Perfect squares are numbers like 4, 9, 16, and 25, which are results of a number multiplied by itself. In our case, the factor of 5 is being squared, so it can be fully simplified when calculating the square root.

Square Root Properties

The properties of square roots are crucial to simplifying expressions, especially when numbers under the square root sign can be broken into perfect squares.

• Basic Property: The square root of a product, \(\sqrt{a \times b}\), can be split into \(\sqrt{a} \times \sqrt{b}\). For example, \(\sqrt{75}\) becomes \(\sqrt{3 \times 5^2}\), which simplifies to \(5\sqrt{3}\).
• Entire vs. Mixed Radicals: Entire radicals look like \(\sqrt{x}\), but many radicals can be split. Mixed radicals show part outside the square root, simplifying the reading and solving process, just like \(5\sqrt{3}\).

Using these properties enables you to simplify radicals efficiently, which is particularly valuable when adding or subtracting them. Recognizing and applying the square root to perfect squares (like \(5^2\)) helps reduce the complexity of expressions.

Radical Expressions

Radical expressions are expressions that contain a square root, cube root, or other roots. Simplifying radical expressions often involves converting them into their most reduced form, using principles like prime factorization and square root properties.

• Simplification: When you encounter a radical expression like \(\sqrt{75}\), use the techniques discussed to simplify it to \(5\sqrt{3}\).
• Combining Like Terms: Just like with regular algebraic expressions, radical expressions can be combined only if they have the same radicand. In the example \(5\sqrt{3} + 5\sqrt{5}\), although they both contain 5, the radicands are different, so they remain separate.

Understanding and simplifying radical expressions is integral to solving many math problems. It helps make calculations easier and results clearer, significantly when combining or subtracting different radicals.