2025年中考數(shù)學(xué)根式試卷及答案

一、單項(xiàng)選擇題（每題2分，共10題）1.若\(\sqrt{a}=2\)，則\(a\)的值是A.4B.-4C.2D.-2答案：A2.下列根式中，最簡(jiǎn)二次根式是A.\(\sqrt{12}\)B.\(\sqrt{18}\)C.\(\sqrt{27}\)D.\(\sqrt{50}\)答案：D3.計(jì)算\(\sqrt{16}+\sqrt{9}\)的結(jié)果是A.5B.7C.8D.10答案：A4.若\(x=\sqrt{3}+1\)，則\(x^2-2x\)的值是A.1B.2C.3D.4答案：B5.下列等式中，正確的是A.\(\sqrt{4+9}=\sqrt{4}+\sqrt{9}\)B.\(\sqrt{4\times9}=\sqrt{4}\times\sqrt{9}\)C.\(\sqrt{16}=4\times2\)D.\(\sqrt{36}=6\)答案：B6.若\(\sqrt{a}-\sqrt=1\)，則\(\sqrt{a^2-b^2}\)的值是A.1B.2C.\(a+b\)D.\(a-b\)答案：A7.計(jì)算\(\sqrt{5}\times\sqrt{20}\)的結(jié)果是A.\(\sqrt{25}\)B.\(\sqrt{100}\)C.\(2\sqrt{5}\)D.\(5\sqrt{4}\)答案：B8.若\(\sqrt{a}+\sqrt=3\)且\(a\)和\(b\)是正數(shù)，則\(\sqrt{a}\cdot\sqrt\)的最大值是A.2B.3C.4D.9答案：C9.下列根式中，有理化分母后結(jié)果是\(3-\sqrt{2}\)的是A.\(\frac{1}{3+\sqrt{2}}\)B.\(\frac{1}{3-\sqrt{2}}\)C.\(\frac{\sqrt{2}}{3+\sqrt{2}}\)D.\(\frac{\sqrt{2}}{3-\sqrt{2}}\)答案：A10.若\(\sqrt{x}=\sqrt{a}+\sqrt\)，則\(x\)的最小值是A.\(a+b\)B.\(a^2+b^2\)C.\(2\sqrt{ab}\)D.\(\sqrt{a^2+b^2}\)答案：A二、多項(xiàng)選擇題（每題2分，共10題）1.下列根式中，屬于最簡(jiǎn)二次根式的是A.\(\sqrt{8}\)B.\(\sqrt{20}\)C.\(\sqrt{50}\)D.\(\sqrt{72}\)答案：C2.下列等式中，正確的是A.\(\sqrt{a^2}=a\)B.\(\sqrt{a^2+b^2}=a+b\)C.\(\sqrt{a^2\cdotb^2}=a\cdotb\)D.\(\sqrt{a^2\cdotb}=a\cdot\sqrt\)答案：C,D3.計(jì)算\(\sqrt{27}\div\sqrt{3}\)的結(jié)果是A.3B.9C.\(\sqrt{3}\)D.\(3\sqrt{3}\)答案：A4.下列根式中，有理化分母后結(jié)果是整數(shù)的是A.\(\frac{1}{\sqrt{2}+\sqrt{3}}\)B.\(\frac{1}{\sqrt{5}-\sqrt{3}}\)C.\(\frac{1}{\sqrt{6}+\sqrt{2}}\)D.\(\frac{1}{\sqrt{7}-\sqrt{1}}\)答案：D5.若\(\sqrt{a}-\sqrt=1\)，則下列等式中正確的是A.\(a-b=1\)B.\(a+b=1\)C.\(a^2-b^2=1\)D.\(a^2+b^2=1\)答案：A,C6.計(jì)算\(\sqrt{50}+\sqrt{32}\)的結(jié)果是A.\(3\sqrt{2}\)B.\(4\sqrt{5}\)C.\(6\sqrt{2}\)D.\(8\sqrt{2}\)答案：C7.下列根式中，屬于無(wú)理數(shù)的是A.\(\sqrt{4}\)B.\(\sqrt{9}\)C.\(\sqrt{10}\)D.\(\sqrt{16}\)答案：C8.若\(\sqrt{a}+\sqrt=\sqrt{c}\)，則\(a+b\)的最小值是A.1B.2C.\(c\)D.\(c^2\)答案：B9.下列根式中，有理化分母后結(jié)果是整數(shù)的是A.\(\frac{1}{\sqrt{3}+\sqrt{2}}\)B.\(\frac{1}{\sqrt{5}-\sqrt{3}}\)C.\(\frac{1}{\sqrt{6}+\sqrt{2}}\)D.\(\frac{1}{\sqrt{7}-\sqrt{1}}\)答案：D10.計(jì)算\(\sqrt{18}\times\sqrt{12}\)的結(jié)果是A.\(6\sqrt{3}\)B.\(12\sqrt{3}\)C.\(36\)D.\(24\sqrt{2}\)答案：B三、判斷題（每題2分，共10題）1.\(\sqrt{25}=5\)答案：正確2.\(\sqrt{a^2+b^2}=a+b\)答案：錯(cuò)誤3.\(\sqrt{16}=-4\)答案：錯(cuò)誤4.\(\sqrt{a}\cdot\sqrt=\sqrt{a\cdotb}\)答案：正確5.\(\sqrt{a}+\sqrt=\sqrt{a+b}\)答案：錯(cuò)誤6.\(\sqrt{27}\div\sqrt{3}=3\)答案：正確7.\(\sqrt{a}-\sqrt=\sqrt{a-b}\)答案：錯(cuò)誤8.\(\sqrt{50}=5\sqrt{2}\)答案：正確9.\(\sqrt{a}\cdot\sqrt=\sqrt{a\cdotb}\)僅當(dāng)\(a\)和\(b\)都為非負(fù)數(shù)時(shí)成立答案：錯(cuò)誤10.\(\sqrt{a^2}=a\)僅當(dāng)\(a\geq0\)時(shí)成立答案：正確四、簡(jiǎn)答題（每題5分，共4題）1.計(jì)算\(\sqrt{45}-\sqrt{20}\)答案：\(\sqrt{45}=3\sqrt{5}\)，\(\sqrt{20}=2\sqrt{5}\)，所以\(\sqrt{45}-\sqrt{20}=3\sqrt{5}-2\sqrt{5}=\sqrt{5}\)2.計(jì)算\(\sqrt{12}\times\sqrt{3}\)答案：\(\sqrt{12}=2\sqrt{3}\)，所以\(\sqrt{12}\times\sqrt{3}=2\sqrt{3}\times\sqrt{3}=2\times3=6\)3.計(jì)算\(\frac{1}{\sqrt{3}+\sqrt{2}}\)答案：有理化分母，\(\frac{1}{\sqrt{3}+\sqrt{2}}\times\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}}=\frac{\sqrt{3}-\sqrt{2}}{3-2}=\sqrt{3}-\sqrt{2}\)4.計(jì)算\(\sqrt{a^2-6a+9}\)答案：\(\sqrt{a^2-6a+9}=\sqrt{(a-3)^2}=|a-3|\)五、討論題（每題5分，共4題）1.討論根式\(\sqrt{a}\)的性質(zhì)答案：根式\(\sqrt{a}\)表示非負(fù)數(shù)的平方根，當(dāng)\(a\geq0\)時(shí)，\(\sqrt{a}\)是實(shí)數(shù)；當(dāng)\(a<0\)時(shí)，\(\sqrt{a}\)無(wú)意義。根式具有非負(fù)性，即\(\sqrt{a}\geq0\)。2.討論根式的運(yùn)算性質(zhì)答案：根式具有乘法、除法、加法、減法等運(yùn)算性質(zhì)。乘法性質(zhì)：\(\sqrt{a}\cdot\sqrt=\sqrt{a\cdotb}\)，除法性質(zhì)：\(\sqrt{a}\div\sqrt=\sqrt{\frac{a}}\)，加法性質(zhì)：\(\sqrt{a}+\sqrt\)不能簡(jiǎn)化為\(\sqrt{a+b}\)，減法性質(zhì)：\(\sqrt{a}-\sqrt\)不能簡(jiǎn)化為\(\sqrt{a-b}\)。3.討論根式的化簡(jiǎn)方法答案：根式的化簡(jiǎn)方法包括有理化分母、化簡(jiǎn)二次根式等。有理