指数与根式化简练习题

复习并运用根式与指数运算法则来化简指数与根式表达式；本页提供详细解答（页面下方）与解析。建议尽量不使用计算器化简数值表达式（仅用于验证答案）。本网站还包含更多关于根式化简的练习题。

含指数的数值表达式化简

第一题

计算下列表达式：
  1. \( 3^2 \)
  2. \( -3^4 \)
  3. \( (-3)^4 \)
  4. \( 12^0 \)
  5. \( 0^{20} \)
  6. \( 0^{-4} \)
  7. \( (-1)^7 \)
  8. \( (-1)^4 \)
  9. \( (-1)^{-3} \)
  10. \( \left( -\dfrac{2}{3} \right)^{-2} \)
  11. \( \dfrac{(-2)^{-2}}{4^{-2}} \)
  12. \( -3^{-3} + (-2)^{-2} \)
  13. \( x^4 \)（当 \( x = -2 \) 时）
  14. \( -x^2 \)（当 \( x = -3 \) 时）
  15. \( -x^5 \)（当 \( x = -2 \) 时）
  16. \( x^4 \)（当 \( x = -\dfrac{2}{3} \) 时）
  17. \( -x^{-2} \)（当 \( x = -\dfrac{1}{2} \) 时）
  18. \( -x^5 \)（当 \( x = -\dfrac{1}{3} \) 时）

含根式与有理指数的数值表达式化简

第二题

计算下列表达式：
  1. \( \sqrt{64} \)
  2. \( \sqrt[3]{-8} \)
  3. \( 4^{3/2} \)
  4. \( 8^{2/3} \)
  5. \( 0.0001^{3/4} \)
  6. \( 16^{-3/4} \)
  7. \( \sqrt{2} \sqrt{8} \)
  8. \( \sqrt[3]{2} \sqrt[3]{32} \)
  9. \( \dfrac{\sqrt{2}}{\sqrt{8}} \)
  10. \( \dfrac{\sqrt[3]{-16}}{\sqrt[3]{2}} \)
  11. \( -2 (\sqrt[6]{8})(5\sqrt[6]{8}) \)
  12. \( \dfrac{-4 \sqrt[3]{375}}{2\sqrt[3]{3}} \)

含指数的代数表达式化简

第三题

化简下列表达式：
  1. \( x^2 \cdot x^5 \)
  2. \( \dfrac{y^4}{y^2} \)
  3. \( (x^2)^{-2} \)
  4. \( 3(\dfrac{1}{2} x^4)(\dfrac{1}{3} x^3) \)
  5. \( (2 x)^4(\dfrac{1}{4} x^{-3}) \)
  6. \( \dfrac{(3x)^2(-2x)^3}{(2x)^2} \)
  7. \( \dfrac{(4x)^2(100 x)^0}{(3x)^2} \)
  8. \( (-2x^2 y^{-3})^3 \)
  9. \( (3x^2 y^3)(2x^5 y^{-2}) \)
  10. \( (-2x^2 y^3 z^4)(-4x^3 y z^{-8})(5x y^2 z^2) \)
  11. \( (-2xy^2)^2 \left (\dfrac{x^6}{(2x)^2} \right)^3 \)
  12. \( \left (\dfrac{4 x^6 y^3}{-8x^3y^{-2}} \right) \)
  13. \( \left (\dfrac{- 4 x^3 y^2}{3 x^3 y^3} \right) \left (\dfrac{3 x^2 y^5}{-6x^3y^2} \right) \)
  14. \( \dfrac{1}{6}(x^2 y z^3)\left (\dfrac{4 x^2 y^3}{3 x^2 y^{-3}z^2} \right) \left (\dfrac{3 x^2 y^5 z}{-x^3 y^2 z^3} \right) \)
  15. \( (4 x^{2/3})(-8x^{1/2}) \)
  16. \( (4 x)^{3/2}(9x)^{1/2} \)
  17. \( (- 27 x^{3/2})^{1/3} \)
  18. \( \left (\dfrac{-8x^3}{y^{-6}} \right)^{2/3} \)
  19. \( x^{5/3} x^{1/6} x ^{-11/6} \)

含根式的代数表达式化简

第四题

  1. \( \sqrt{x^2} \)
  2. \( \sqrt[4]{16x^4} \)
  3. \( \sqrt{(2x - 1)^2} \)
  4. \( \sqrt[3]{-27x^3} \)
  5. \( \sqrt[3]{8 x^6 y^3} \)
  6. \( \sqrt{x^3} \sqrt{x^2} \)
  7. \( \sqrt[3]{\left (\dfrac{-8x^6}{y^{-3}} \right)} y \)
  8. \( \dfrac{ \sqrt[5]{64x^9 y^7}}{ \sqrt[5]{2 x^4 y^2}} \)
  9. \( (4\sqrt[8]{b^2})( 5\sqrt[8]{b^3})( \sqrt[8]{b^3}) \)

习题解答

第一题解答

计算下列表达式：
  1. \(3^2 = 3 \times 3 = 9\)
  2. \(-3^4 = - (3 \times 3 \times 3 \times 3) = -81\)
  3. \((-3)^4 = -3 \times -3 \times -3 \times -3 = 81\)
  4. \(12^0 = 1\)
  5. \(0^{20} = 0\)
  6. \(0^{-4} = \dfrac{1}{0^4} = \dfrac{1}{0}\)：未定义，因为数学中不允许除以零。
  7. \((-1)^7 = -1\)
  8. \((-1)^4 = 1\)
  9. \((-1)^{-3} = \dfrac{1}{(-1)^3} = \dfrac{1}{-1} = -1\)
  10. \(\left(-\dfrac{2}{3}\right)^{-2} = \left(-\dfrac{3}{2}\right)^2 = (-1)^2 \left(\dfrac{3}{2}\right)^2 = \dfrac{9}{4}\)
  11. \(\dfrac{(-2)^{-2}}{4^{-2}} = \dfrac{4^2}{(-2)^2} = \dfrac{16}{4} = 4\)
  12. \(-3^{-3} + (-2)^{-2} = -\dfrac{1}{3^3} + \dfrac{1}{(-2)^2} = -\dfrac{1}{27} + \dfrac{1}{4} = \dfrac{23}{108}\)
  13. \(x^4\)（当 \(x = -2\) 时） \[ x^4 = (-2)^4 = 16 \]
  14. \(-x^2\)（当 \(x = -3\) 时） \[ - x^2 = -(-3)^2 = -9 \]
  15. \(-x^5\)（当 \(x = -2\) 时） \[ - x^5 = -(-2)^5 = 32 \]
  16. \(x^4\)（当 \(x = -\dfrac{2}{3}\) 时） \[ \left(-\dfrac{2}{3}\right)^4 = \dfrac{(-2)^4}{3^4} = \dfrac{16}{81} \]
  17. \(-x^{-2}\)（当 \(x = -\dfrac{1}{2}\) 时） \[ - x^{-2} = -\left(-\dfrac{1}{2}\right)^{-2} = -(-1)^{-2} \left(\dfrac{1}{2}\right)^{-2} \] \[ = -1 \cdot \left(\dfrac{2}{1}\right)^2 = -\dfrac{4}{1} = -4 \]
  18. \(-x^5\)（当 \(x = -\dfrac{1}{3}\) 时） \[ - x^5 = -\left(-\dfrac{1}{3}\right)^5 = -(-1)^5 \left(\dfrac{1}{3}\right)^5 \] \[ = -(-1) \cdot \dfrac{1}{243} = \dfrac{1}{243} \]

第二题解答

计算下列表达式：
  1. \(\sqrt{64} = \sqrt{8^2} = 8\)
  2. \(\sqrt[3]{-8} = \sqrt[3]{(-2)^3} = -2\)
  3. \(4^{3/2} = (\sqrt{4})^3 = 2^3 = 8\)
  4. \(8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4\)
  5. \(0.0001^{3/4} = (\sqrt[4]{0.0001})^3\) \[ = (\sqrt[4]{\dfrac{1}{10000}})^3 = (\sqrt[4]{\dfrac{1}{10^4}})^3 \] \[ = \left(\dfrac{1}{10}\right)^3 = \dfrac{1}{10^3} = 0.001 \]
  6. \(16^{-3/4} = \dfrac{1}{16^{3/4}} = \dfrac{1}{(\sqrt[4]{16})^3}\) \[ = \dfrac{1}{2^3} = \dfrac{1}{8} \]
  7. \(\sqrt{2} \sqrt{8} = \sqrt{2 \times 8} = \sqrt{16} = 4\)
  8. \(\sqrt[3]{2} \sqrt[3]{32} = \sqrt[3]{2 \times 32} = \sqrt[3]{64} = 4\)
  9. \(\dfrac{\sqrt{2}}{\sqrt{8}} = \sqrt{\dfrac{2}{8}} = \sqrt{\dfrac{1}{4}} = \dfrac{1}{2}\)
  10. \(\dfrac{\sqrt[3]{-16}}{\sqrt[3]{2}} = \sqrt[3]{\dfrac{-16}{2}} = \sqrt[3]{-8} = -2\)
  11. \(-2 (\sqrt[6]{8})(5\sqrt[6]{8}) = (-2 \times 5)\sqrt[6]{8 \times 8}\) \[ = -10\sqrt[6]{64} = -10\sqrt[6]{2^6}\] \[ = -10(2) = -20 \]
  12. \(\dfrac{-4\sqrt[3]{375}}{2\sqrt[3]{3}} = \left(\dfrac{-4}{2}\right)\sqrt[3]{\dfrac{375}{3}} = -2\sqrt[3]{125}\) \[ = -2 \times 5 = -10 \]

第三题解答

化简下列表达式：
  1. \( x^2 \cdot x^5 = x^{2 + 5} = x^7 \)
  2. \( \frac{y^4}{y^2} = y^{4 - 2} = y^2 \)
  3. \( (x^2)^{-2} = x^{2 \times (-2)} = x^{-4} = \frac{1}{x^4} \)
  4. \(3(\dfrac{1}{2} x^4)(\dfrac{1}{3} x^3) \) \[ = (3 \times \dfrac{1}{2} \times \dfrac{1}{3} ) x^{4 + 3} = \dfrac{1}{2} x^7 \]
  5. \( (2 x)^4(\dfrac{1}{4} x^{-3}) = 2^4 x^4 (\dfrac{1}{4} x^{-3}) \) \[ = (2^4 \times \dfrac{1}{4}) x^{4-3} = 4 x \]
  6. \( \dfrac{(3x)^2(-2x)^3}{(2x)^2} = \dfrac{3^2 x^2 (-2)^3 x^3}{2^2 x^2} \) \[ = \dfrac{3^2 (-2)^3}{2^2} \dfrac{x^3}{x^2} = - 18 x \]
  7. \( \dfrac{(4x)^2(100 x)^0}{(3x)^2} = \dfrac{4^2 x^2 \times 1}{3^2 x^2} = 16/9 \)
  8. \( (-2x^2 y^{-3})^3 = (-2)^3 (x^2)^3 (y^{-3})^3 \) \[ = -8 x^6 y^{-9} = \dfrac{-8x^6}{y^9} \]
  9. \(3x^2 y^3)(2x^5 y^{-2}) = (3 \times 2)(x^2 x^5)(y^3 y^{-2}) \) \[ = 6x^{2 + 5} y^{3 - 2} = 6x^7 y \]
  10. \( (-2x^2 y^3 z^4)(-4x^3 y z^{-8})(5x y^2 z^2) \) \[ = (-2 \times -4 \times 5)(x^2 x^3 x)(y^3 y y^2)(z^4 z^{-8} z^2) \] \( = 40x^{2 + 3 + 1} y^{3 + 1 + 2} z^{4 - 8 + 2} \) \[ = 40x^6 y^6 z^{-2} = \dfrac{40x^6 y^6}{z^2} \]
  11. \( (-2xy^2)^2 \left (\dfrac{x^6}{(2x)^2} \right)^3 \) \[ = (-2)^2 x^2 y^4 \left (\dfrac{x^{18}}{(2x)^6} \right) = \dfrac{4x^2 y^4x^{18}}{2^6 x^6} \] \[ = \dfrac{4}{2^6} \dfrac{x^{2+18}y^4}{x^6} = \dfrac{1}{16} {x^{14} y^4} \]
  12. \( \left (\dfrac{4 x^6 y^3}{-8x^3y^{-2}} \right) \) \[ = \dfrac{4}{-8} \; \dfrac{x^6 y^3}{x^3y^{-2}} \] \[ = -\dfrac{1}{2} x^{6-3} y^{3-(-2)} = -\dfrac{1}{2} x^3 y^5 \]
  13. \( \left (\dfrac{- 4 x^3 y^2}{3 x^3 y^3} \right) \left (\dfrac{3 x^2 y^5}{-6x^3y^2} \right) \) \[ = \dfrac{(- 4 \times 3)}{(3 \times (-6))} \dfrac{(x^3 y^2)(x^2 y^5)}{(x^3 y^3)(x^3y^2)} \] \[ = \dfrac{2}{3} \dfrac{x^5 y^7}{x^6 y^5} = \dfrac{2}{3} x^{5-6} y^{7-5}= \dfrac{2}{3} \dfrac{y^2}{x} \]
  14. \( \dfrac{1}{6}(x^2 y z^3)\left (\dfrac{4 x^2 y^3}{3 x^2 y^{-3}z^2} \right) \left (\dfrac{3 x^2 y^5 z}{-x^3 y^2 z^3} \right) \) \[ = \dfrac{1 \times 4 \times 3}{6 \times 3 \times (-1)} \dfrac{(x^2 y z^3)(x^2 y^3)(x^2 y^5 z)}{(x^2 y^{-3}z^2)(-x^3 y^2 z^3)} \] \[ = (-\dfrac{2}{3}) \dfrac{x^{2+2+2}y^{1+3+5}z^{3+1}}{x^{2+3} y^{-3+2} z^{2+3} } \] \[ = - \dfrac{2}{3} \dfrac{x^6 y^9 z^4}{x^5 y z^5} = - \dfrac{2}{3} \; \dfrac{x y^{10} }{z} \]
  15. \( (4 x^{2/3})(-8x^{1/2}) = (4 \times (-8))(x^{2/3+1/2}) = - 32 x^{7/6} \)
  16. \( (4 x)^{3/2}(9x)^{1/2} = 4^{3/2} x^{3/2} 9^{1/2} x^{1/2} \) \[ = (\sqrt 4)^3 (\sqrt 9) x^{3/2+1/2} = 8 \times 3 x^2 = 24 x^2 \]
  17. \( (- 27 x^{3/2})^{1/3} = (- 27)^{1/3} (x^{3/2})^{1/3} \) \[ = \sqrt[3]{-27} x^{3/2 \times 1/3} = - 3 x^{1/2} = - 3 \sqrt x \]
  18. \( \left (\dfrac{-8x^3}{y^{-6}} \right)^{2/3} = \dfrac{(-8x^3)^{2/3}} {(y^{-6})^{2/3}} \) \[ = \dfrac{(-8)^{2/3} (x^3)^{2/3}} {(y^{-6 \times (2/3)}} \] \[ = \dfrac{(\sqrt[3]{-8})^2 (x^{3 \times (2/3)})} {(y^{-4})} = 4 x^2 y^4 \]
  19. \( x^{5/3} x^{1/6} x ^{-11/6} = x^{5/3+1/6 -11/6} = x^0 = 1 , x \ne 0 \)

第四题解答

  1. \( \sqrt{x^2} = | x | \)
  2. \( \sqrt[4]{16x^4} = \sqrt[4]{16} \sqrt[4]{x^4} = 4 | x | \)
  3. \( \sqrt{(2x - 1)^2} = |2x - 1| \)
  4. \( \sqrt[3]{-27x^3} = \sqrt[3]{-27} \sqrt[3]{x^3} = \sqrt[3]{(-3)^3} \sqrt[3]{x^3} = - 3 x \)
  5. \( \sqrt[3]{8 x^6 y^3} = \sqrt[3]{8} \sqrt[3]{ x^6} \sqrt[3]{y^3} = 2 x^{6/3} y = 2 x^2 y \)
  6. \( \sqrt{x^3} \sqrt{x^2} = \sqrt{x^3 x^2} = \sqrt{x^{3+2}} = \sqrt{x^5} = \sqrt{x^4 \times x} = x^2 \sqrt x \)
  7. \( \sqrt[3]{\left (\dfrac{-8x^6}{y^{-3}} \right)} = \dfrac{\sqrt[3]{-8x^6}}{\sqrt[3]{y^{-3}}} \) \[ = \dfrac{\sqrt[3]{-8} \sqrt[3]{x^6}} {\sqrt[3]{y^{-3} }} \] \[ = \dfrac{\sqrt[3]{(-2)^3} \sqrt[3]{(x^2)^3}} {\sqrt[3]{(y^{-1})^3 }} = \dfrac{-2 x^2}{y^{-1}} = -2 x^2 y \]
  8. \( \dfrac{ \sqrt[5]{64x^9 y^7}}{ \sqrt[5]{2 x^4 y^2}} = \sqrt[5] {\dfrac{64x^9 y^7}{2 x^4 y^2}}\) \[ = \sqrt[5] {32 x^{9-4} y^{7-2}} = \sqrt[5] {2^5} \sqrt[5] x^{5} \sqrt[5] y^{5} = 2 x y \]
  9. \( (4\sqrt[8]{b^2})( 5\sqrt[8]{b^3})( \sqrt[8]{b^3}) \) \[ = (4 \times 5) \sqrt[8]{b^2 b^3 b^3 } = 20 \sqrt[8]{b^8 } = 20 |b| = 20 b \] 注意：最后一题中，由于 \( \sqrt[8]{b^3}\) 是实数，因此 b 大于或等于零。故 |b| = b。

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