20 道二次根式乘法计算题（含答案与解析）

20道二次根式乘法计算题（含答案与解析）以下题目均遵循二次根式乘法法则：\(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\)（\(a\geq0\)，\(b\geq0\)），计算后需将结果化为最简二次根式（被开方数不含分母、不含能开得尽方的因数或因式）。一、基础型（直接应用乘法法则）题目：\(\sqrt{2}\times\sqrt{3}\)解析：直接套用法则，\(\sqrt{2}\times\sqrt{3}=\sqrt{2\times3}=\sqrt{6}\)（\(\sqrt{6}\)已是最简形式）答案：\(\sqrt{6}\)题目：\(\sqrt{5}\times\sqrt{5}\)解析：\(\sqrt{5}\times\sqrt{5}=\sqrt{5\times5}=\sqrt{25}=5\)（\(\sqrt{25}\)开方得整数）答案：\(5\)题目：\(\sqrt{6}\times\sqrt{12}\)解析：先算乘积，\(\sqrt{6\times12}=\sqrt{72}\)；再化简，\(\sqrt{72}=\sqrt{36\times2}=\sqrt{36}\times\sqrt{2}=6\sqrt{2}\)答案：\(6\sqrt{2}\)题目：\(\sqrt{8}\times\sqrt{2}\)解析：\(\sqrt{8\times2}=\sqrt{16}=4\)（\(\sqrt{16}\)是完全平方数，直接开方）答案：\(4\)题目：\(\sqrt{10}\times\sqrt{15}\)解析：\(\sqrt{10\times15}=\sqrt{150}=\sqrt{25\times6}=5\sqrt{6}\)答案：\(5\sqrt{6}\)二、含系数型（系数与根式分别相乘）题目：\(2\sqrt{3}\times3\sqrt{2}\)解析：系数相乘（\(2\times3\)），根式相乘（\(\sqrt{3}\times\sqrt{2}\)），再合并结果：\(2\sqrt{3}\times3\sqrt{2}=(2\times3)\times(\sqrt{3}\times\sqrt{2})=6\sqrt{6}\)答案：\(6\sqrt{6}\)题目：\(5\sqrt{6}\times2\sqrt{3}\)解析：系数：\(5\times2=10\)；根式：\(\sqrt{6\times3}=\sqrt{18}=3\sqrt{2}\)；整体：\(10\times3\sqrt{2}=30\sqrt{2}\)答案：\(30\sqrt{2}\)题目：\(\sqrt{2}\times4\sqrt{8}\)解析：系数：\(1\times4=4\)；根式：\(\sqrt{2\times8}=\sqrt{16}=4\)；整体：\(4\times4=16\)答案：\(16\)题目：\(3\sqrt{5}\times\sqrt{5}\)解析：系数：\(3\times1=3\)；根式：\(\sqrt{5\times5}=5\)；整体：\(3\times5=15\)答案：\(15\)题目：\(2\sqrt{12}\times3\sqrt{3}\)解析：先化简\(\sqrt{12}=2\sqrt{3}\)，再计算：\(2\times2\sqrt{3}\times3\sqrt{3}=4\sqrt{3}\times3\sqrt{3}=(4\times3)\times(\sqrt{3}\times\sqrt{3})=12\times3=36\)答案：\(36\)三、被开方数含分数型（先去分母，再化简）题目：\(\sqrt{\frac{1}{2}}\times\sqrt{8}\)解析：法则适用分数，\(\sqrt{\frac{1}{2}\times8}=\sqrt{4}=2\)（或先有理化\(\sqrt{\frac{1}{2}}=\frac{\sqrt{2}}{2}\)，再算\(\frac{\sqrt{2}}{2}\times2\sqrt{2}=\frac{2\times2}{2}=2\)）答案：\(2\)题目：\(\sqrt{\frac{3}{4}}\times\sqrt{12}\)解析：\(\sqrt{\frac{3}{4}\times12}=\sqrt{9}=3\)（或\(\frac{\sqrt{3}}{2}\times2\sqrt{3}=\sqrt{3}\times\sqrt{3}=3\)）答案：\(3\)题目：\(\sqrt{\frac{2}{5}}\times\sqrt{\frac{5}{8}}\)解析：\(\sqrt{\frac{2}{5}\times\frac{5}{8}}=\sqrt{\frac{10}{40}}=\sqrt{\frac{1}{4}}=\frac{1}{2}\)（分子分母约分后再开方）答案：\(\frac{1}{2}\)题目：\(2\sqrt{\frac{1}{3}}\times\sqrt{27}\)解析：\(2\times\sqrt{\frac{1}{3}\times27}=2\times\sqrt{9}=2\times3=6\)答案：\(6\)四、混合运算型（结合加法或多步乘法）题目：\(\sqrt{3}\times(\sqrt{6}+\sqrt{3})\)解析：用乘法分配律：\(\sqrt{3}\times\sqrt{6}+\sqrt{3}\times\sqrt{3}=\sqrt{18}+3=3\sqrt{2}+3\)答案：\(3+3\sqrt{2}\)题目：\((\sqrt{2}+\sqrt{5})\times\sqrt{2}\)解析：分配律展开：\(\sqrt{2}\times\sqrt{2}+\sqrt{5}\times\sqrt{2}=2+\sqrt{10}\)答案：\(2+\sqrt{10}\)题目：\(\sqrt{2}\times\sqrt{3}\times\sqrt{6}\)解析：分步计算：\(\sqrt{2\times3}\times\sqrt{6}=\sqrt{6}\times\sqrt{6}=6\)（或\(\sqrt{2\times3\times6}=\sqrt{36}=6\)）答案：\(6\)题目：\(3\sqrt{2}\times2\sqrt{5}\times\sqrt{10}\)解析：系数：\(3\times2=6\)；根式：\(\sqrt{2\times5\times10}=\sqrt{100}=10\)；整体：\(6\times10=60\)答案：\(60\)五、拓展型（被开方数含字母，需注意取值范围）题目：\(\sqrt{x}\times\sqrt{x^3}\)（\(x\geq0\)）解析：\(\sqrt{x\timesx^3}=\sqrt{x^4}=x^2\)（\(x\geq0\)时，\(\sqrt{x^4}=(x^2)^{\frac{1}{2}\times2}=x^2\)）答案：\(x^2\)题目：\(2\sqrt{xy}\times3\sqrt{x^2y}\)（\(x\geq0\)，\(y\geq0\)）解析：系数：\(2\times3=6\)；根式：\(\sqrt{xy\timesx^2y}=\sqrt{x^3y^2}=xy\sqrt{x}\)（\(\sqrt{x^3}=x\sqrt{x}\)，\(\sqrt{y^2}=y\)）；整体：\(6\timesxy\sqrt{x}=6xy\sqrt{x}\)答案：\(6xy\sqrt{x}\)核对提示若结果含根号，需确认被开方数