Simplify each radical. See Example 5. √75

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Identify the number inside the square root: here it is 75.

Factor 75 into its prime factors or into a product of a perfect square and another number. For example, 75 can be factored as \(75 = 25 \times 3\).

Use the property of square roots that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\) to rewrite \(\sqrt{75}\) as \(\sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3}\).

Simplify the square root of the perfect square: \(\sqrt{25} = 5\).

Write the simplified form as \(5 \sqrt{3}\), which is the simplified radical form of \(\sqrt{75}\).

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Simplifying Radicals

Simplifying radicals involves expressing a square root in its simplest form by factoring the radicand into perfect squares and other factors. For example, √75 can be broken down into √(25 × 3), which simplifies to 5√3 because √25 is 5.

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Simplifying Trig Expressions

Perfect Squares

Perfect squares are numbers that are squares of integers, such as 1, 4, 9, 16, 25, etc. Recognizing perfect squares within a radicand helps simplify radicals by extracting these square roots as whole numbers.

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Solving Quadratic Equations by Completing the Square

Properties of Square Roots

The property √(a × b) = √a × √b allows the separation of a square root of a product into the product of square roots. This property is essential for breaking down complex radicals into simpler parts for easier simplification.

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