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Chapter 8: Radicals 8.1 Radicals - Square Roots Square roots are the most common type of radical used. A square root “un-squares” a number. For example, because 52 = 25 we say the square root of 25 is 5. The square root of 25 is written as 25 √. The following example gives several square roots: Example 1. 1 √ =1 121 √ = 11 4 √ =2 ...
More directly, when determining a product or quotient of radicals and the indices (the small number in front of the radical) are the same then you can rewrite 2 radicals as 1 or 1 radical as 2. Simplify by rewriting the following using only one radical sign (i.e. rewriting 2 radicals as 1).
Understanding radicals is essential in solving quadratics using the quadratic formula and the distance formula. If x is a number greater than or equal to zero, x represents the positive square root of x and − x represents the negative square root of x. To understand the terminology of radicals, study the illustration below.
A radical expression is an expression that contains a radical. An expression involving a radical with index n is in simplest form when these three conditions are met: • No radicands have perfect nth powers as factors other than 1. • No radicands contain fractions. • No radicals appear in the denominator of a fraction.
In this section you will study roots and see how powers and roots are related. We use the idea of roots to reverse powers. Because 32 9 and ( 3)2 9, both 3 and 3 are square roots of 9. Because 24 16 and ( 2)4 16, both 2 and 2 are fourth roots of 16. Because 23 8 and ( 2)3 8, there is only one real cube root of 8 and only one real cube root of 8.
A radical expression is an expression that contains a radical. An expression involving a radical with index n is in simplest form when these three conditions are met. • No radicands have perfect nth powers as factors other than 1. • No radicands contain fractions. • No radicals appear in the denominator of a fraction.
Convert between radical notation and exponential notation. Simplify expressions with rational exponents using the properties of exponents. Multiply and divide radical expressions with different indices. Solve equations with radicals and check for extraneous solutions.
