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

Verified step by step guidance

1. First, simplify each square root term by factoring the number inside the root into a product of a perfect square and another factor. For example, express 32 as 16 \(\times\) 2 and 98 as 49 \(\times\) 2.
2. Rewrite each term using the property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). So, \( \sqrt{32} = \sqrt{16} \times \sqrt{2} \) and \( \sqrt{98} = \sqrt{49} \times \sqrt{2} \).
3. Calculate the square roots of the perfect squares: \( \sqrt{16} = 4 \) and \( \sqrt{49} = 7 \). Substitute these back into the expression.
4. Rewrite the original expression with the simplified roots: \( -5 \times 4 \times \sqrt{2} + 2 \times 7 \times \sqrt{2} \).
5. Factor out the common radical \( \sqrt{2} \) and combine the coefficients: \( (-5 \times 4 + 2 \times 7) \times \sqrt{2} \). This will give you the simplified expression.

Key Concepts

Simplifying Radicals

Simplifying radicals involves expressing the number under the square root as a product of perfect squares and other factors. This allows you to extract the square root of the perfect square, making the radical simpler. For example, √32 can be written as √(16×2) = 4√2.

Like Radicals

Like radicals have the same radicand (the number inside the root) and the same index. Only like radicals can be added or subtracted directly by combining their coefficients. For instance, 4√2 and 2√2 are like radicals and can be combined as (4+2)√2 = 6√2.

Adding and Subtracting Radicals

To add or subtract radicals, first simplify each radical to identify like terms. Then, combine the coefficients of like radicals by addition or subtraction. If radicals are not like terms, they remain separate in the expression.