Math topic中文

Laws of indices

laws of indices

Practice questions

  1. 1计算： (1)(94)12−(−9.6)0−(278)−23+(32)−2(1)(\dfrac{9}{4})^{\frac{1}{2}}-(-9.6)^{0}-(\dfrac{27}{8})^{-\frac{2}{3}}+(\dfrac{3}{2})^{-2}； (2)lg⁡5+lg⁡2⋅lg⁡5+(lg⁡2)2+eln⁡3(e(2)\lg 5+\lg 2\boldsymbol{⋅}\lg 5+(\lg 2)^{2}+e^{\ln 3}(e…
  2. 2下列算式中，其中正确的有()①(−12)−3=−18()①(- \dfrac {1}{2})^{-3}=- \dfrac {1}{8}，②a2+2a−1=(a−1)2②a^{2}+2a-1=(a-1)^{2}，③a8÷a8=1(a≠0)③a^{8}÷a^{8}=1(a\neq 0)，④(a−b)2=a2−b2④(a-b)^{2}=a^{2}-b^{2}…
  3. 3下列计算正确的是(    )(\:\:\:\:) A. a2+a3=a5a ^{2} +a ^{3} =a ^{5} B. a3⋅a3=a9a ^{3} \boldsymbol{⋅}a ^{3} =a ^{9} C. (a3)2=a6(a ^{3} ) ^{2} =a ^{6}…
  4. 4Express (x3)2(\sqrt[3]{x})^2 in index notation. Answer ... Find the integer nn such that 2n=1322^n = \dfrac{1}{32}. Answer n=n =…
  5. 5下列运箅正确的是()() A. 2a2−a=a2a^{2}-a=a B. (a+2)2=a2+4(a+2)^{2}=a^{2}+4 C. (a2)3=a6(a^{2})^{3}=a^{6} D. (−3)2=−3 \sqrt {(-3)^{2}}=-3
  6. 6Simplify 3a25b3+(12c25ab)2\frac{3a^2}{5b^3} + \left(\frac{12c}{25ab}\right)^2. nn is a positive integer. Show that, for all nn, (3n+2)2−(3n−2)2(3n+2)^2 - (3n-2)^2…
  7. 7下列各式中正确的是()(\quad) A. a4⋅a2=a8a^{4}\cdot a^{2}=a^{8} B. (−2a2)3=−6a6(-2a^{2})^{3}=-6a^{6} C. 6a2+13a=2a+1\dfrac{6a^{2}+1}{3a}=2a+1 D. a(a−3)=a2−3aa(a-3)=a^{2}-3a…
  8. 8Written as a product of its prime factors, p=33×5×72p = 3^3 \times 5 \times 7^2. Express 450 as a product of its prime factors.…
  9. 9若4x−3y−3=04x-3y-3=0，则104x÷103y=10^{4x}÷10^{3y}=___.
  10. 10Given that 464x=1\frac{4}{64^x} = 1, find the value of x. Answer x = ___
  11. 11已知数列{an}\{a_{n}\}的前nn项和为SnS_{n}，首项为a1a_{1}，且44，ana_{n}，SnS_{n}成等差数列． (1)求数列{an}\{a_{n}\}的通项公式； (2)若an2=2bna_{n}^{2}={{2}^{{{b}_{n}}}}…
  12. 12Write 92x+19^{2x+1} as a power of 3. Simplify 9a24b(27a3b6)−1\dfrac{9a^2}{4b}\left(\dfrac{27a^3b}{6}\right)^{-1}.
  13. 13化简求值： (1)0.064−13−(18)0+1634+0.2512(2)12lg⁡25+lg⁡2+(13)log⁡32−log⁡29×log⁡32.(1)0.{064}^{- \frac{1}{3}}-( \dfrac{1}{8}{)}^{0}+{16}^{ \frac{3}{4}}+0.{25}^{ \frac{1}{2}} (2)\dfrac{1}{2} \lg 25+\lg 2+( \dfrac{1}{3}{)}^{{\log }_{3}2} -\log _{2}9×\log _{3}2.…
  14. 14下列计算正确的是()(\quad) A. x2+x2=x4x^{2}+x^{2}=x^{4} B. x2⋅x3=x5x^{2}\boldsymbol{⋅}x^{3}=x^{5} C. (xy)3=x3y(xy)^{3}=x^{3}y D. (x4)3=x7(x^{4})^{3}=x^{7}…
  15. 15下列函数中，满足f(x2)=[f(x)]2f(x^{2})=[f(x)]^{2}的是()() A. f(x)=ln⁡xf(x)=\ln x B. f(x)=∣x+1∣f(x)=|x+1| C. f(x)=x3f(x)=x^{3} D. f(x)=exf(x)=e^{x}
  16. 16Solve the equation 33x×125x=2253^{3x} \times 125^{x} = 225.
  17. 17Solve the equation 3−2x7=13(2x−1).\displaystyle \frac{3-2x}{7} = \frac{1}{3}(2x-1). Answer x = ... Simplify 25a2−9b215a−10ab+6b2−9b\displaystyle \frac{25a^{2}-9b^{2}}{15a-10ab+6b^{2}-9b}…
  18. 18If 10m=210^{m}=2 and 10n=310^{n}=3, find the value of 103m−n10^{3m-n}. Answer ...
  19. 19下面计算正确的是（ ） A. x3÷x3=0 B. x3-x2=x C. x3·x2=x6 D. x3÷x3=x
  20. 20下列运算正确的是()() A. x3−3x2=−2xx^{3}-3x^{2}=-2x B. (−13x3)2=19x6(- \dfrac {1}{3}x^{3})^{2}= \dfrac {1}{9}x^{6} C. 6x3÷2x−2=3x6x^{3}÷2x^{-2}=3x D.…
  21. 21Solve the inequality 4x+13>3−2x5\frac{4x+1}{3} > \frac{3-2x}{5}. Simplify (16a12b8)−14\left(\dfrac{16a^{12}}{b^8}\right)^{-\tfrac{1}{4}}, l…
  22. 22Solve the inequality −3x+42<5−x7\frac{-3x+4}{2} < \frac{5-x}{7}. Answer ... Simplify (64a6b−15)−13\left(\dfrac{64a^{6}}{b^{-15}}\right)^{-\tfrac{1}{3}}…
  23. 23Simplify (x327y6)43\left(\frac{x^3}{27y^6}\right)^{\frac{4}{3}}.
  24. 24如果3a3b2÷13ab=A3a^{3}b^{2}÷\dfrac{1}{3}ab=A，那么AA是()(\quad) A. a3b4a^{3}b^{4} B. a2ba^{2}b C. 3a4b33a^{4}b^{3} D. 9a2b9a^{2}b
  25. 25下列运算正确的是()() A. a10+a5=a2a^{10}+a^{5}=a^{2} B. (a3)4=a7(a^{3})^{4}=a^{7} C. (x−y)2=x2−y2(x-y)^{2}=x^{2}-y^{2} D. x3⋅(−x3)=−x6x^{3}⋅(-x^{3})=-x^{6}…
  26. 26Simplify Simplify 5−2(6x−1)5-2(6x-1). Simplify (27b9a6)−1/3\left(\dfrac{27b^9}{a^6}\right)^{-1/3}.
  27. 27计算(−3a2)2(-3a^{2})^{2}的结果是()() A. 3a43a^{4} B. −3a4-3a^{4} C. 9a49a^{4} D. −9a4-9a^{4}
  28. 28计算：0.25×(−12)−4−4÷(5−1)0−(116)−12=0.25×(- \dfrac {1}{2})^{-4}-4÷( \sqrt {5}-1)^{0}-( \dfrac {1}{16})^{- \frac {1}{2}}= ___ ．
  29. 29Simplify (2x−122x)2\left(\dfrac{2^{x-1}\sqrt{2}}{2^x}\right)^2. Answer ... If 27m3x=127m^{3x}=1 and m>0m>0, find m2x+m−xm^{2x}+m^{-x}. Answ…
  30. 30计算(a3b)2(a^{3}b)^{2}的结果是()() A. a6ba^{6}b B. a6b2a^{6}b^{2} C. a5b2a^{5}b^{2} D. a3b2a^{3}b^{2}
  31. 31The first four terms in a sequence of numbers are given below. Find T5T_5. Answer ... Find an expression, in terms of n,…
  32. 32Write down, and simplify the terms in the expansion of (2+x)5(2 + x)^5 in ascending powers of xx. Hence find the coefficien…
  33. 33下列计算正确的是()(\quad) A. (a3)2=a5(a^{3})^{2}=a^{5} B. (2a)2=2a3(2a)^{2}=2a^{3} C. a⋅a3=a4a\boldsymbol{⋅}a^{3}=a^{4} D. 2a−a=22a-a=2
  34. 34T=2πLgT = 2\pi\sqrt{\dfrac{L}{g}} Rearrange the formula to make g the subject. Answer g = ...
  35. 35Given that 21003+21000=2n×3m2^{1003} + 2^{1000} = 2^n \times 3^m, find the value of mm and of nn. Ans: m = ___, n = ___
  36. 36Solve the equation (15)−2×125x=25x(\frac{1}{5})^{-2}\times125^{x}=\sqrt{25^{x}}. Answer x = ...
  37. 37计算： (1)(12)−1−(−8)0+22018×(−12)2017(2)(2a2)2⋅a4−(−5a4)2(1){{(\dfrac{1}{2})}^{-1}}-{{(-8)}^{0}}+{{2}^{2018}}\times {{(-\dfrac{1}{2})}^{2017}}(2){{(2{{a}^{2}})}^{2}}\cdot {{a}^{4}}-{{(-5{{a}^{4}})}^{2}}…
  38. 38计算(−0.25)2021×(−4)2020(-0.25)^{2021}×(-4)^{2020}的结果是()(\quad) A. −14-\dfrac{1}{4} B. 14\dfrac{1}{4} C. −4-4 D. 44
  39. 39Given that 12a−1=23+22+23+23\frac{1}{2^{a-1}} = 2^3 + 2^2 + 2^3 + 2^3, find the value of a.
  40. 40Given that (17)3y=735\left(\frac{1}{7}\right)^{3y} = 7^{\frac{3}{5}}, find the value of y. Ans: y = ___
  41. 41下列运算中，正确的是()() A. x2⋅x3=x6x^{2}⋅x^{3}=x^{6} B. (x3)2=x5(x^{3})^{2}=x^{5} C. x+x2=2x3x+x^{2}=2x^{3} D. −x3÷x2=−x-x^{3}÷x^{2}=-x
  42. 42计算：(π−5)0+4+(−1)2015−3tan⁡60∘(π- \sqrt {5})^{0}+ \sqrt {4}+(-1)^{2015}- \sqrt {3}\tan 60^{\circ}．
  43. 43下列运算正确的是( ) A. 2x+2x=4x22x+2x=4{{x}^{2}} B. x8÷x2=x4{{x}^{8}}\div {{x}^{2}}={{x}^{4}} C. x2⋅x3=x5{{x}^{2}}\cdot {{x}^{3}}={{x}^{5}} D…
  44. 44The CoV (coronavirus) is circular in shape with a diameter of approximately 0.00014 mm. Express 0.00014 in standard form…
  45. 45下列计算正确的是()(\quad) A. x5−x3=x2x^{5}-x^{3}=x^{2} B. (x+2)2=x2+4(x+2)^{2}=x^{2}+4 C. (−2x2)3=−8x6(-2x^{2})^{3}=-8x^{6} D. x2⋅x3=x6x^{2}\boldsymbol{⋅}x^{3}=x^{6}…
  46. 46The number of bacteria in a colony doubles every hour. The colony starts with: 50 bacteria. The table below shows the nu…
  47. 47Given that 25×125x=525\times125^{x}=5, find the value of xx. Answer x=x= ...
  48. 48用科学记数法表示下列各数： (1)405.6=(1)405.6=;(2)10000=;(2)10000=;(3)65000000=;(3)65000000=;(4)3386×1013=;(4)3386\times 10^{13}= .
  49. 49The prime factorisation of 450 is 2×32×522 \times 3^2 \times 5^2. Express 648 as a product of its prime factors. Answer 648 =…
  50. 50计算(x2)3×(−2x)4(x^{2})^{3}×(-2x)^{4}的结果是()() A. 16x916x^{9} B. 16x1016x^{10} C. 16x1216x^{12} D. 16x2416x^{24}
  51. 51Write as a single fraction in its simplest form. 3t2w⋅9t2w3\dfrac{3t^2}{w} \cdot \dfrac{9t^2}{w^3} Answer ----------------------…
  52. 52Simplify. a33a2×27a6\dfrac{a^3}{3a^2}\times 27a^6 Answer ... 416x3\sqrt{\dfrac{4}{16x^3}} Answer ... 492a7b=3434c\dfrac{49^{2a}}{7^b}=343^{4c}…
  53. 53Find the value of 5+265−26\sqrt{\dfrac{5+2\sqrt{6}}{5-2\sqrt{6}}}, giving your answer in the form a+b6a + b\sqrt{6}. Without usi…
  54. 54Express 9801 as a product of its prime factors. Answer --------------------------------------------------------- Hence,…
  55. 55Simplify (16y3)32(16y^3)^{\frac{3}{2}}. 5k=1255355^k = 125\sqrt[3]{5}\sqrt{5} Use laws of indices to find the value of kk. Show your…
  56. 56下列运算正确的是()(\quad) A. a3⋅a4=a12a^{3}\boldsymbol{⋅}a^{4}=a^{12} B. (m3)2=m5(m^{3})^{2}=m^{5} C. x3+x3=x6x^{3}+x^{3}=x^{6} D. (−a2)3=−a6(-a^{2})^{3}=-a^{6}…
  57. 57The sketch shows the graph of y=ka−xy = ka^{-x}. The points A (−1, 6) and B (0, 3) lie on the graph. Find the value of k and…
  58. 58Sketch the graph of y=2ln⁡xy = 2\ln x. In order to solve the equation xe2x=13xx e^{2x} = \dfrac{1}{3x}, a suitable straight line ha…
  59. 59计算： -28x4y2÷7x3y＝
  60. 60已知a+2b=2a+2b=2，则4a+16b4^{a}+16^{b}的最小值为()() A. 22 B. 44 C. 88 D. 1616