Problem 39 - Algebra 2

Question

Simplify. \(\sqrt{98}-\sqrt{72}+\sqrt{32}\)

Step-by-Step Solution

Answer

\(5\sqrt{2}\)

Step 1: Factor Each Radicand

For each square root, factor the number inside the square root (known as the radicand) into its prime factors. - **98**: The prime factorization is \(2 \times 7^2\). - **72**: The prime factorization is \(2^3 \times 3^2\). - **32**: The prime factorization is \(2^5\).

Step 2: Simplify Each Square Root

Use the formula \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\), and simplify each square root using the prime factorization.- **\(\sqrt{98}\)**: Simplifies to \(\sqrt{2} \times \sqrt{7^2} = 7\sqrt{2}\).- **\(\sqrt{72}\)**: Simplifies to \(\sqrt{2^3} \times \sqrt{3^2} = 6\sqrt{2}\).- **\(\sqrt{32}\)**: Simplifies to \(\sqrt{2^4} \times \sqrt{2} = 4\sqrt{2}\).

Step 3: Combine Like Terms

Now, combine the terms that have the same radical part (\(\sqrt{2}\) in this case).Combine: \(7\sqrt{2} - 6\sqrt{2} + 4\sqrt{2} = (7 - 6 + 4)\sqrt{2} = 5\sqrt{2}\).

Step 4: Write the Final Simplified Expression

The expression simplifies to \(5\sqrt{2}\). This is the final simplified form.

Key Concepts

Prime Factorization

Prime factorization is a critical technique used in simplifying radicals. Let's start with a quick understanding of what it means. When you break down a composite number into a product of smaller prime numbers, you perform prime factorization. Why use prime factorization?- It's useful because it's a common method to simplify square roots by identifying perfect squares.- Knowing the factors helps in taking out numbers easily from under the square root sign.For our problem, you factor each radicand (the number inside the square root) into its prime constituents:

• 98: Prime factors are \(2 \times 7^2\).
• 72: Prime factors are \(2^3 \times 3^2\).
• 32: Prime factors are \(2^5\).

After finding these factors, the next step is using them to simplify the square roots, which involves the perfect squares you just identified.

Combine Like Terms

Now that you have simplified each square root using prime factorization, the process of combining like terms becomes essential. Combining like terms refers to grouping together terms that share the same variable or radical component. In this problem, all terms contain \(\sqrt{2}\). Here's what you do:

• Identify the common radical, which is \(\sqrt{2}\) here.
• Write each term as a product of its coefficient and the common radical, e.g., \(7\sqrt{2}\), \(6\sqrt{2}\), and \(4\sqrt{2}\).
• Add or subtract the coefficients (7, -6, and 4 in this case). You combine them like this: \((7 - 6 + 4)\sqrt{2} = 5\sqrt{2}\).

This is a straightforward way to simplify expressions involving radicals with similar components, making the final expression clearer and simpler to evaluate.

Square Roots

Understanding square roots is essential when simplifying radicals. A square root asks the question: "What number, when multiplied by itself, gives me the original number?"Let’s delve into the actual simplification process. With square roots, you're often looking for perfect squares. These are numbers that have whole numbers as their square roots, like 1, 4, 9, 16, and so on.Here's how to simplify:

• If the radicand can be expressed as a product with a perfect square, you can "take out" the square root of that perfect square.
• For example, with \(\sqrt{98}\), notice it's \(\sqrt{2 \times 7^2}\). You can "take out" 7, simplifying it to \(7\sqrt{2}\).
• This process helps reduce the complexity of the squared terms and makes them easier to combine or manipulate.

This simplification is essential to reduce your problem to its simplest form, ensuring that all square roots remain manageable and clearly defined.