Add or subtract, as indicated. See Example 6. √45 + 4√20

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Start by simplifying each square root term separately. Recall that \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \), and look for perfect squares inside the radicals.

Simplify \( \sqrt{45} \) by expressing 45 as \( 9 \times 5 \), so \( \sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5} \).

Simplify \( 4\sqrt{20} \) by first simplifying \( \sqrt{20} \). Express 20 as \( 4 \times 5 \), so \( \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5} \). Then multiply by 4 to get \( 4 \times 2\sqrt{5} = 8\sqrt{5} \).

Now rewrite the original expression using the simplified terms: \( 3\sqrt{5} + 8\sqrt{5} \).

Since both terms have the common radical \( \sqrt{5} \), combine the coefficients: \( (3 + 8)\sqrt{5} = 11\sqrt{5} \).

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Simplifying Radicals

Simplifying radicals involves expressing a square root in its simplest form by factoring out perfect squares. For example, √45 can be rewritten as √(9×5) = 3√5. This process makes it easier 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). Only like radicals can be added or subtracted directly by combining their coefficients. For instance, 3√5 and 4√5 can be added to get 7√5.

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Rationalizing Denominators

Adding and Subtracting Radical Expressions

To add or subtract radicals, first simplify each radical and identify like radicals. Then, combine the coefficients of like radicals while keeping the radical part unchanged. This is similar to combining like terms in algebra.

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