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The side length of a square is the square root of its area. This relationship is shown by a radical symbol . The number or expression under the radical symbol is called the radicand. The radical symbol indicates only the positive square root of a number, called the principal root. To indicate both the positive and negative square roots of a number, use the plus or minus sign (±). or –5 Numbers such as 25 that have integer square roots are called perfect squares. Square roots of integers that are not perfect squares are irrational numbers. You can estimate the value of these square roots by comparing them with perfect squares. For example, lies between and , so it lies between 2 and 3. Ex 1: Estimate to the nearest tenth. < < 5< <6 Find the two perfect squares that 27 lies between. Find the two integers that lies between . Because 27 is closer to 25 than to 36, Try 5.2: 5.22 = 27.04 5.12 = 26.01 Too high, try 5.1. Too low Because 27 is closer to 27.04 than 26.01, than to 5.1. Check On a calculator to the nearest tenth. is close to 5 than to 6. is closer to 5.2 ≈ 5.1961524 ≈ 5.1 rounded Notice that these properties can be used to combine quantities under the radical symbol or separate them for the purpose of simplifying square-root expressions. A square-root expression is in simplest form when the radicand has no perfect-square factors (except 1) and there are no radicals in the denominator. Ex 2: Simplify each expression. A. B. Find a perfect square factor of 32. Product Property of Square Roots Product Property of Square Roots D. C. Quotient Property of Square Roots Quotient Property of Square Roots If a fraction has a denominator that is a square root, you can simplify it by rationalizing the denominator. To do this, multiply both the numerator and denominator by a number that produces a perfect square under the radical sign in the denominator. Ex 3: Simplify by rationalizing the denominator. B. A. Multiply by a form of 1. =2 Multiply by a form of 1. Square roots that have the same radicand are called like radical terms. To add or subtract square roots, first simplify each radical term and then combine like radical terms by adding or subtracting their coefficients. Ex 4: Adding and Subtracting Square Roots A. B. Simplify radical terms. Combine like radical terms.