Add or subtract, as indicated. See Example 6.-5√32 + 2√98

Verified step by step guidance

1. Step 1: Simplify each square root term separately. Start with \( \sqrt{32} \).
2. Step 2: Express \( \sqrt{32} \) as \( \sqrt{16 \times 2} \), which simplifies to \( \sqrt{16} \times \sqrt{2} = 4\sqrt{2} \).
3. Step 3: Simplify \( \sqrt{98} \) by expressing it as \( \sqrt{49 \times 2} \), which simplifies to \( \sqrt{49} \times \sqrt{2} = 7\sqrt{2} \).
4. Step 4: Substitute the simplified forms back into the expression: \( -5\sqrt{32} + 2\sqrt{98} \) becomes \( -5(4\sqrt{2}) + 2(7\sqrt{2}) \).
5. Step 5: Combine like terms by factoring out \( \sqrt{2} \): \( (-5 \times 4 + 2 \times 7)\sqrt{2} \).

Key Concepts

Simplifying Radicals

Simplifying radicals involves reducing square roots to their simplest form by factoring out perfect squares. For example, √32 can be simplified by recognizing that 32 = 16 × 2, leading to √32 = √(16 × 2) = 4√2. This process is essential for combining like terms in expressions involving square roots.

Combining Like Terms

Combining like terms is a fundamental algebraic process where terms with the same variable or radical part are added or subtracted. In the expression -5√32 + 2√98, after simplifying each radical, you can combine the coefficients of like radicals to arrive at a final simplified expression. This step is crucial for obtaining a concise answer.

Arithmetic with Radicals

Arithmetic with radicals requires understanding how to perform addition and subtraction with terms that include square roots. When adding or subtracting radicals, only like radicals can be combined, meaning they must have the same radicand. This concept is vital for correctly manipulating expressions that involve square roots.