ALGEBRA - solved problems Radicals (roots) and/or exponentiation with fractional (rational) exponent
 67 Find given roots or radicals.
 Solutions: a) since  33 = 27. b) since  (-3)3 = (-3)·(-3)·(-3) = - 27. c) since  (-2)5 = (-2)·(-2)·(-2)·(-2)·(-2) = -32.
The odd root of any real number exists.
 68 Find given roots.
 Solutions: a) since  42 = 16 and  4 > 0,   [recall that  ( ± 4)2 = 16]. b) since  34 = 81 and  3 > 0,   [recall that  ( ± 3)4 = 81].
The even root of a nonnegative real number exists. Note that the even root is defined to be a nonnegative number.
The even root of a negative real number does not exist as a real number.
 For example, do not exist as real numbers, but they do exist as complex numbers.
Rules and properties of radicals and/or fractional exponents
 If m, n and p are natural numbers (n >1) and if  a and b are nonnegative real numbers, then Properties Examples                Solutions:    Operations on radical expressions
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