Add or subtract, as indicated. See Example 6. 2√50 - 5√72

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Start by simplifying each square root term separately. Recall that \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \), where \(a\) is a perfect square if possible.

Simplify \( 2\sqrt{50} \) by expressing 50 as \( 25 \times 2 \), so \( 2\sqrt{50} = 2 \times \sqrt{25 \times 2} = 2 \times \sqrt{25} \times \sqrt{2} \).

Simplify \( 5\sqrt{72} \) by expressing 72 as \( 36 \times 2 \), so \( 5\sqrt{72} = 5 \times \sqrt{36 \times 2} = 5 \times \sqrt{36} \times \sqrt{2} \).

Calculate the square roots of the perfect squares: \( \sqrt{25} = 5 \) and \( \sqrt{36} = 6 \), then rewrite the terms as multiples of \( \sqrt{2} \).

Now that both terms have the common radical \( \sqrt{2} \), combine them by subtracting the coefficients: \( (2 \times 5)\sqrt{2} - (5 \times 6)\sqrt{2} = (10 - 30)\sqrt{2} \).

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Simplifying Radicals

Simplifying radicals involves expressing the square root of a number as a product of simpler square roots, often by factoring out perfect squares. For example, √50 can be simplified to 5√2 because 50 = 25 × 2 and √25 = 5. This step is essential to combine like terms in radical expressions.

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Simplifying Trig Expressions

Like Radicals

Like radicals have the same radicand (the number inside the square root) and can be added or subtracted by combining their coefficients. For instance, 3√2 and 5√2 are like radicals and can be combined as (3 + 5)√2 = 8√2. Identifying like radicals is crucial for performing addition or subtraction.

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Rationalizing Denominators

Arithmetic Operations with Radicals

Adding or subtracting radicals requires first simplifying them and then combining like terms by adding or subtracting their coefficients. Unlike regular numbers, radicals can only be combined if their radicands match. This concept ensures correct manipulation of expressions involving square roots.

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