Simplifying radicals: Section 2We 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 . . Therefore in general we must write . conforms to the definition of the absolute value. Example 4. Compare ()² and . ()² = x. (Lesson 26.) For, in order for that radical to be a real number, the radicand x may not be negative. = x -- only if x ≥ 0. For any value of x, we must write . Example 5. Since the square of any power produces an even exponent -- (a^{3})^{2} = a^{6} -- then the square root of an even power will be half the exponent. = a^{3}. As for an odd power, such as a^{7}, it is composed of an even power times a: a^{7} = a^{6}a. Therefore, = = a^{3}. (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.
Note: '2n' in algebra, as in part c), indicates an even number, that is, a multiple of 2. The variable n typically signifies an integer. We signify an odd number, then, as '2n + 1,' as in part g). 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 rule of algebra? (Assume that a and b do not have negative values.)
Problem 8. Express each radical in simplest form. a) = = 2. To simplify a radical, the radicand must be composed of factors! b) = = 2a c) = = 3b Fractional radicand A radical is in its simplest form when the radicand is not a fraction. Example 6. 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 9. Simplify . Solution. The denominator must be a perfect square. We can make 50 into a square number simply by multiplying by 2. We can make x a square by multiplying by x. And y² is already a square. Therefore, Example 10. Simplify . (Assume that the variables do not have negative values.) Solution. Again, 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.)
A problem that asks you to show, means to write what's on the left, and then transform it algebraically so that it looks like what's on the right.
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