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　　GRE数学知识点之数的乘方及开方介绍

　　一、基本概念

　　1. Powers of Numbers (乘方)：这种求n个相同因数的积的运算，叫做乘方，乘方的结果叫做幂。在an 中，a叫做底数，n叫做指数，an读作a的n次方。an看作是a的n次方的结果时，也可读作a的n次幂。二次方也叫平方，三次方也叫立方。

　　正数的任何次幂都是正数;负数的奇次幂是负数，负数的偶次幂是正数。

　　2. Roots of Numbers (开方)：一般地，如果一个数的平方等于a，这个数就叫做a的平方根(也叫做二次 方根)，换句话说，如果x2=a，则x就叫做a的平方根。

　　一般来说，一个正数有两个平方根，这两个平方根互为相反数，零的平方根为零。在式子士

　　中，a叫做被开方数，2叫做根指数。

　　正数a的正的平方根，也叫做a的算术平方根;零的平方根也叫做零的算术平方根，因此零的算术平方根仍旧为零。

　　二、幂的基本性质

　　三、Exponents，Roots and Real Number Line(幂、方根与实数轴)

　　1. Exponents and the Real Number Line(幂和实数轴)

　　Raising numbers to powers can have surprising effects on the size and/or sign (negative vs. positive) of the base number. This is one of the test-makers’ favorite areas! The impact of raising a number to an exponent (power) depends on the region on the number line where the number and exponent fall. Here are the four regions you need to consider：

　　(1) Less than —1 (to the left of —1 on the number line)

　　(2) Between —1 and 0

　　(3) Between 0 and 1

　　(4) Greater than 1 (to the right of 1 on the number line)

　　2. Roots and the Real Number Line (平方根及实数轴)

　　As with exponents, the root of a number can bear a surprising relationship to the size and/or sign (negative vs. positive ) of the number (another favorite area of the test-makers). Here are our observations you should remember.

　　(1) If n>1, then

　　(the higher the roots, the lower the value). However, if n lies between 0 and 1，then

　　(the higher the root, the higher the value).

　　(2) The square root of any negative number is an imaginary number(虚数)，not a real number. Remember: you won’t encounter imaginary numbers on the GRE.

　　(3) Every positive number has two square roots： a negative number and a positive number (with the same absolute value ). The same holds true for all other even-numbered roots of positive numbers.

　　(4) Every negative number has exactly one cube root，and that root is a negative number. The same holds true for all other odd-numbered roots of negative numbers.

　　(5) Every positive number has only one cube root, and that root is always a positive number. The same holds true for all other odd-numbered roots of positive numbers.

　　四、The Operation Rule of Radicals(根式的运算法则)

　　1. Combining Radicals (根式的合并)

　　(1) Addition and Subtraction(加法与减法)：If a term under a radical is being added to or subtracted from a term under a different radical，you cannot combine the two terms under the same radical.(注：根式相加减时，不能把不同根式下的数直接相加减。)

　　(2) Multiplication and Division(乘法与除法)：Terms under different radicals can be combined under a common radical if one term is multiplied or divided by the other, but only if the root is the same.(注：仅当两根式的幂指数相同时，才能把它们的根式的下面的数相乘除。)

　　2. Simplifying Radicals (化简根式)：

　　On the GRE, always look for the possibility of simplifying radicals by moving part of what’s inside the radical to the outside. Check inside your square-root radicals for factors that are squares of nice tidy numbers (especially integers).

　　注：在GRE考试中，如根式下面有可提取到根式外面去的因子，一定要把该因子提到根式的外面去，若 分母上有根式时，一般要先把分母上的根式约去，这样可以使运算的步骤大为简化。

　　GRE数学知识点之数的乘方及开方实例讲解

　　If —1

　　本题的正确答案为(E)。五个选项从小到大的顺序依次是：(E)，(B)，(A)，(D)，(C)。其分析如下所示：

　　(C) equals 1 (any non-zero term raised to the power of zero equals 1)

　　(D) is a positive number between 0 and 1

　　(A) is a positive number between 0 and |x| , which is the value of (D)

　　(B) is a negative non-integer between 0 and x

　　(E) is a negative number less than (to the left of ) —1

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