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
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 = 26, which equation correctly expresses P in terms of C and V?
原库参考解析
根式与有理指数挑战
\( \sqrt{x^2-20x+100}-11x=0 \)
What are all possible solutions to the given equation?
原库参考解析
根式与有理指数进阶
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 = 28, which equation correctly expresses \( P \) in terms of \( C \) and \( V \) ?
原库参考解析
根式与有理指数挑战
\( \sqrt{\left(x-70\right)^2}=-6x \)
If \( x \) \( < \) 0, what is the value of \( x \) ?
原库参考解析
根式与有理指数进阶
b2+5c=9dThe given equation relates the real numbers b, c, and d, where d > 5/9c. Which equation correctly expresses b in terms of c and d?
原库参考解析
根式与有理指数进阶
b2 +5c =9d
The given equation relates the real numbers b, c, and d, where d >\( \frac{5}{9} \)c. Which equation correctly express b in terms of c and d ?
原库参考解析
根式与有理指数进阶
If \( x^n=\sqrt{2x^n+35} \) , where n is a positive integer constant, what is the value of xn?
原库参考解析
根式与有理指数挑战
\( \sqrt{k-x\ }=39-x \)
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
根式与有理指数挑战
Which expression is equivalent to \( \sqrt{x^{197}} \) , where \( x>0 \) ?
原库参考解析
根式与有理指数进阶
\( \sqrt{2x+19}=5 \)
What is the solution to the equation shown?
原库参考解析
根式与有理指数进阶
The function \( f \) is defined by \( f\left(x\right)=4\sqrt{x} \). For what value of x does \( f\left(x\right)=28 \)?
原库参考解析
根式与有理指数挑战
\( \sqrt{\left(x+15\right)^2}=x+15 \)
What is the least value of x that is a solution to the given equation?
原库参考解析
