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一个非负数的非负方根,叫做这个数的算术方根,简称算术根。因此,当且仅当a≥0时,根式a~(1/n)(n∈N且n≥2)表示a的n次算术根。在实数范围内,当n为偶数且a<0时。a~(1/n)无意义;而n为奇数,a<0时,a~(1/n)虽然有意义,但它不是算术根。对此,学生容易搞错。因为根式的运算法则都是针对算术根而言,所以把一个非算术根化为算术根就显得十分重要。例如,a~(1/(2n-1))(a<0,n∈N且n≥2)化成-(-a)~(1/(2n-1))或-|a|~(1/(2n-1)),这里(-a)~(1/(2n-1))或|a|~(1/(2n-1))就是算术根了。一般的,分指数幂都限制其底数大于零。即是说,一个根式化为分指数幂,也是立足于算术根的。它的意义是:a~(m/n)=a~(m/n)(a≥0,m、n∈N且n≥2)。由于学生对算术根和分指数幂的规定含糊不清,导至根式或分指数幂运算的错误的例证是不胜枚举的。就
A non-negative, non-negative root, called the arithmetic root of this number, is referred to as the arithmetic root. Therefore, if and only if a ≥ 0, the root a(1/n) (n∈N and n≥2) represents the n-th arithmetic root of a. In the real range, when n is even and a<0. a~(1/n) is meaningless; n is odd, and a~(1/n) is not an arithmetic root, though it is significant. In this regard, students are prone to mistakes. Because the root algorithm is for the arithmetic root, it is important to root a non-arithmetic to an arithmetic root. For example, a~(1/(2n-1)) (a<0, n∈N and n≥2) is converted into -(-a)~(1/(2n-1)) or -|a|~(1 /(2n-1)), where (-a)~(1/(2n-1)) or |a|~(1/(2n-1)) is the arithmetic root. In general, sub-exponential powers limit the base number to greater than zero. That is, one rooted into a sub-exponential power is also based on an arithmetic root. Its significance is: a~(m/n)=a~(m/n) (a≥0, m, n∈N and n≥2). Since students’ ambiguities about the arithmetic roots and subexponential powers are ambiguous, examples of errors that lead to radical or sub- exponentiation are numerous. on