Simplifying radicals: Section 2 (If you are not viewing this page with Internet Explorer 6, then your browser may not be able to display the symbol ≥, "is greater than or equal to;" or ≤, "is less than or equal to.") We can identify with the absolute value of x (Lesson 12). . For, when x ≥ 0, then . But if x < 0 -- if, for example, x = −5 -- then
-- because the square root is never negative. (Lesson 26.) Rather, when x < 0, then . . This conforms to the definition of the absolute value. Simplifying powers Example 4. Since the square of any power produces an even exponent - (a3)² = a6 -- then the square root of an even power will be half the exponent. = a3 As for an odd power, such as a7, it is composed of an even power times a: a7 = a6a Therefore, = = a3. (These results hold only for a ≥ 0.) Problem 5. Simplify each radical. (Assume a ≥ 0.) To see the answer, pass your mouse over the colored area.
Problem 6. Simplify each radical. Remove the even powers. (Assume that the variables do not have negative values.)
Factors of the radicand Problem 7. True or false? That is, which of these is a formal rule of algebra? (Assume that the variables do not have negative values.)
Problem 8. Express each radical in simplest form. a) = = 2. Make factors! b) = = 2a c) = = 3b Fractional radicand A radical is also in it simplest form when the radicand is not a fraction. Example 5. The denominator a square number. When the denominator is a square number, as , then
In general, For, a· a = a².
Problem 9. Simplify each radical.
Solution. When the denominator is not a square number, we can make it a square number by multiplying it. In this example, we will multiply it by itself, that is, by 2. But then we must multiply the numerator also by 2:
Example 8. Simplify . (Assume that the variables do not have negative values.) Solution. The denominator must be a perfect square. It must be composed of even powers. Therefore, make the denominator into a product of even powers simply simply by multiplying it -- and the numerator -- by bc. Then extract half of the even powers. Problem 10. Simplify each radical. (Assume that the variables do not have negative values.)
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