Ch. 1 - Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problem 74

Chapter 2, Problem 74

Exercises 73–75 will help you prepare for the material covered in the next section. Simplify: √18 - √8

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Identify the radicals to simplify: \( \sqrt{18} - \sqrt{8} \).

Express each radicand as a product of a perfect square and another factor: \( 18 = 9 \times 2 \) and \( 8 = 4 \times 2 \).

Rewrite each square root using the property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \): \( \sqrt{18} = \sqrt{9} \times \sqrt{2} \) and \( \sqrt{8} = \sqrt{4} \times \sqrt{2} \).

Simplify the square roots of the perfect squares: \( \sqrt{9} = 3 \) and \( \sqrt{4} = 2 \), so the expression becomes \( 3\sqrt{2} - 2\sqrt{2} \).

Combine like terms by subtracting the coefficients of \( \sqrt{2} \): \( (3 - 2)\sqrt{2} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Simplifying Square Roots

Simplifying square roots involves expressing the radicand as a product of perfect squares and other factors. For example, √18 can be rewritten as √(9×2), which simplifies to 3√2 because √9 is 3. This process makes it easier to combine or subtract roots.

Like Radicals

Like radicals have the same radicand and index, allowing them to be combined through addition or subtraction. For instance, 3√2 and 5√2 are like radicals and can be combined as (3 - 5)√2 = -2√2. Recognizing like radicals is essential for simplifying expressions involving roots.

Subtraction of Radicals

To subtract radicals, first simplify each radical and then combine like radicals by subtracting their coefficients. If the radicals are not like terms, they cannot be combined further. For example, √18 - √8 simplifies to 3√2 - 2√2, which equals √2.

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