这部分需要会什么？
- 在根式和有理指数之间转换
- 检查实数范围内根式的定义条件
- 解根式方程并检验增根
核心概念与公式
对正数 \(a\)，\(a^{m/n}=\sqrt[n]{a^m}\)。偶次根的被开方数在实数范围内必须非负，偶次根号取非负主值。因此 \(\sqrt{x^2}=|x|\)，不是对所有 \(x\) 都等于 \(x\)。奇次根可以处理负数。
平方可以消去平方根，但也可能引入原式不满足的解。例如 \(\sqrt{A(x)}=B(x)\) 要求 \(A(x)\ge0\) 且 \(B(x)\ge0\)；平方后的方程不能单独保证这些条件。
做题时怎样使用
- 先写被开方数和分母的限制；将根式单独放在一边
- 两边平方或使用指数法则，再求候选解
- 把每个候选解代回原方程，只保留真正成立的解
容易混淆的地方
\(\sqrt{u+v}\) 一般不等于 \(\sqrt u+\sqrt v\)。解 \(x^2=9\) 有正负两个解，但 \(\sqrt9\) 本身等于 3。
用一个例题串起方法
知识点教学例题解 \(\sqrt{x+2}=x\)。
展开解题思路 ＋
右边必须非负，故 \(x\ge0\)。平方得 \(x+2=x^2\)，即 \((x-2)(x+1)=0\)。候选解为 2 和 \(-1\)，其中 \(-1\) 不满足原式。唯一解为 \(x=2\)。
本知识点题库
先独立作答，再查看答案与解析。
根式与有理指数进阶
The function g is defined by \( g\left(x\right)=\frac{\sqrt{cx}}{c}+17 \), where c is a negative constant. What is the value of \( g\left(25c\right) \)?
原库参考解析
根式与有理指数挑战
The function k is defined by \( k\left(s\right)=\sqrt{s+110} \). If \( k\left(53p\right)=p \), where p is a constant, what is the value of p?
原库参考解析
根式与有理指数挑战
If n and k are numbers greater than 1 and \( \sqrt[4]{n^{5}} \) is equivalent to \( \sqrt[5]{k^{2}} \), for what value of a is \( n^{2a+1} \) equal to k?
原库参考解析
根式与有理指数进阶
Which expression is equivalent to\( \left(x\right)^{\frac{1}{17}} \), where \( x>0 \)?
原库参考解析
根式与有理指数挑战
\( r=\sqrt{9w-5} \)
The given equation relates the real numbers r and w, where \( w>\frac{5}{9} \).Which equation correctly expresses w in terms of r ?
原库参考解析
根式与有理指数挑战
For a certain circuit, its power \( P \), in watts, current \( C \), in amperes; voltage \( V \), in volts; and resistance \( R \), in ohms, are related as \( \frac{CV^2}{P}=\sqrt{PR} \), where \( P \), \( C \), \( V \), and \( R \) are positive. When \( R=18 \), which equation correctly expresses \( P \) in terms of \( C \) and \( V \)?
原库参考解析
根式与有理指数进阶
The functions g and h are defined by the given equations.
\( g\left(x\right)=\sqrt{\left(x-11\right)^2+45} \)
\( h(x)=\sqrt[3]{64}+\frac{x^{2}}{2} \)
If \( g\left(8\right)=t \) , where t is a constant, what is the value of \( h\left(t\right) \)?
原库参考解析
根式与有理指数挑战
\( \frac{\sqrt[3]{x^{17}y^{5}}}{5x^{2}(\sqrt[8]{y^{8}})} \)
For all positive values of x and y, the given expression is equivalent to which of the following?
Ⅰ.\( \frac{(x^{\frac{11}{3}})(y^{\frac{2}{3}})}{5} \)
Ⅱ.\( \frac{\sqrt[3]{x^{11}y^{2}}}{5} \)
原库参考解析
根式与有理指数挑战
\( \sqrt[5]{x^{m}} \)
In the given expression, m is a constant and \( x>1. \) The expression can be rewritten as \( \sqrt[7]{x} \). What is the value of m?
原库参考解析
根式与有理指数挑战
\( 49=\frac{2K}{v^2} \)
For an object with a mass of 49 kilograms, the given equation relates the kinetic energy K, in joules, of the object to the object's speed v, in meters per second, where K and v are positive. Which equation correctly expresses this object's speed, in meters per second, in terms of the object's kinetic energy, in joules?
原库参考解析
根式与有理指数挑战
If n and k are numbers greater than 1 and \( \sqrt{n^{5}} \) is equivalent to \( \sqrt[5]{k^{4}} \) , for what value of a is \( n^{4a+1} \) equal to k?
原库参考解析
根式与有理指数挑战
\( \sqrt{k-x}=28-x \)
In the given equation, k is a constant. The equation has exactly one real solution. What is the minimum possible value of 4k?
原库参考解析
