问题9.填写空白：
(i)∛(125×27)= 3×…
(ii)∛(8×…)= 8
(iii)281728 = 4×…
(iv)∛480=∛3×2×∛..
(v)∛…=∛7×∛8
(vi)∛.. =∛4×∛5×∛6
(vii)∛(27/125)=…/ 5
(viii)∛(729/1331)= 9 /…
(ix)∛(512 /…)= 8/13
解决方案：
(i) ∛(125 × 27) = 3 × ….
Let’s solve LHS,
∛(125 × 27) = ∛(5 × 5 × 5 × 3 × 3 × 3)
= 5 × 3
So missing value in RHS is 5
(ii) ∛(8 × …) = 8
Let’s solve LHS,
∛(8 × …) = ∛8 × 8 × 8 = 8
So missing value in LHS is 8
(iii) ∛1728 = 4 × …
Let’s solve LHS,
∛1728 = ∛2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
= 2 × 2 × 3 = 4 × 3
So missing value in RHS is 3
(iv) ∛480 = ∛3×2× ∛..
Let’s solve LHS,
∛480 = ∛2 × 2 × 2 × 2 × 2 × 3 × 5 = 2 × ∛3 × ∛2× 2 × 5
= 2 × ∛3 × ∛20
So missing value in RHS is 20
(v) ∛… = ∛7 × ∛8
Let’s solve RHS,
∛7 × ∛8 = ∛(7 × 8) = ∛56
So missing value in LHS is 56
(vi) ∛..= ∛4 × ∛5 × ∛6
Let’s solve RHS,
∛4 × ∛5 × ∛6 = ∛(4 × 5 × 6) = ∛120
So missing value in LHS is 120
(vii) ∛(27/125) = …/5
Let’s solve LHS,
∛(27/125)= ∛(3 × 3 × 3)/(5 × 5 × 5) = 3/5
So missing value in RHS is 3
(viii) ∛(729/1331) = 9/…
Let’s solve LHS,
∛(729/1331)= ∛(9 × 9 × 9)/(11 × 11 × 11) = 9/11
So missing value in RHS is 11
(ix) ∛(512/…) = 8/13
Let’s solve RHS,
Taking cube and cube root of 8/13 at the same time
So, ∛(8/13)3 = ∛(8 × 8 × 8)/(13 × 13 × 13) = ∛512/2197
So missing value in LHS is 2197
问题10.立方箱的体积为474. 552立方米。找到盒子每一边的长度。
解决方案：
Let a be the length of each side of the box
The volume of cubical box = 474.552 m3
As, we know volume of cube = (side)3
So, a3 = 474552/1000
a = ∛474552/1000
Solving the cube root of 474552/1000
= ∛(2 × 2 × 2 × 3 × 3 × 3 × 13 × 13 × 13)/(10 × 10 × 10)
= 2 × 3 × 13/10 = 7.8 m
Hence, the length of each side of the box is 7.8 m
问题11：三个数字彼此之间是2：3：4。他们的立方体的总和是0.334125。找出数字
解决方案：
Let the three numbers be 2a, 3a, and 4a
It is given that the sum of their cubes is 0.334125
So, 2a3 + 3a3 + 4a3 = 334125/1000000
99a3 = 334125/1000000 or a = ∛334125/(1000000 × 99)
= ∛3375/1000000 = ∛15 × 15 × 15/10 × 10 × 10 × 10 × 10 × 10
= 15/100 = 0.15
So, first number = 2a = 2 × 0.15 = 0.3
The second number = 3a = 3 × 0.15 = 0.45
The third number = 4a = 4 × 0.15 = 0.6
Hence, the numbers are 0.3, 0.45, and 0.6
问题12.找到一个体积为24389 / 216m 3的立方体的边。
解决方案：
Let a be the length of each side of the cube
The volume of cube = 24389/216 m3
As, we know volume of cube = (side)3
So, a3 = 24389/216
a = ∛24389/216
Solving the cube root of 24389/216
= ∛(29 × 29 × 29)/(2 × 2 × 2 × 3 × 3 × 3)
= 29/2 × 3 = 29/6 = 4.84 m
Hence, the length of each side of the cube is 4.84 m
问题13：评估：
(i)∛36×∛384
(ii)∛96×∛144
(iii)∛100×∛270
(iv)∛121×∛297
解决方案 ：
(i) ∛36 × ∛384
We know that, ∛a × ∛b = ∛ab
= ∛(36 × 384) = ∛(2 × 2 × 3 × 3) × (2 × 2 × 2 × 2 × 2 × 2 × 2 × 3)
= ∛23 × 23 × 23 × 33
= 2 × 2 × 2 × 3 = 24
Hence, ∛36 × ∛384 = 24
(ii) ∛96 × ∛144
We know that, ∛a × ∛b = ∛ab
= ∛(96 × 144) = ∛(2 × 2 × 2 × 2 × 2 × 3) × (2 × 2 × 2 × 2 × 3 × 3)
= ∛23 × 23 × 23 × 33
= 2 × 2 × 2 × 3 = 24
Hence, ∛96 × ∛144 = 24
(iii) ∛100 × ∛270
We know that, ∛a × ∛b = ∛ab
= ∛(100 × 270) = ∛27000
= ∛3 × 3 × 3 × 10 × 10 × 10 = 3 × 10 = 30
Hence, ∛100 × ∛270 = 30
(iv) ∛121 × ∛297
We know that, ∛a × ∛b = ∛ab
= ∛(121 × 297) = ∛(11 × 11) × (3 × 3 × 3 × 11)
= ∛33 × 113
= 3 × 11 = 33
Hence, ∛121 × ∛297 = 33
问题14。使用以下事实找到数字2460375、20346417、210644875、57066625的立方根
(i)2460375 = 3375×729
(ii)20346417 = 9261×2197
(iii)210644875 = 42875×4913
(iv)57066625 = 166375×343
解决方案：
(i) 2460375 = 3375 × 729
Cube root of 2460375 will be written as,
∛2460375 = ∛3375 × 729
= ∛(3 × 3 × 3 × 5 × 5 × 5) × (9 × 9 × 9)
= ∛33 × 53 × 93
= 3 × 5 × 9 = 135
Hence, cube root of 2460375 is 135
(ii) 20346417 = 9261 × 2197
Cube root of 20346417 will be written as,
∛20346417 = ∛9261 × 2197
= ∛(3 × 3 × 3 × 7 × 7 × 7) × (13 × 13 × 13)
= ∛33 × 73 × 133
= 3 × 7 × 13 = 273
Hence, cube root of 20346417 is 273
(iii) 210644875 = 42875 × 4913
Cube root of 210644875 will be written as,
∛210644875 = ∛42875 × 4913
= ∛(5 × 5 × 5 × 7 × 7 × 7) × (17 × 17 × 17)
= ∛53 × 73 × 173
= 5 × 7 × 17 = 595
Hence, cube root of 210644875 is 595
(iv) 57066625 = 166375 × 343
Cube root of 57066625 will be written as,
∛57066625 = ∛166375 × 343
= ∛(5 × 5 × 5 × 11 × 11 × 11) × (7 × 7 × 7)
= ∛53 × 113 × 73
= 5 × 11 × 7 = 385
Hence, cube root of 57066625 is 385
问题15.查找以下数字的立方根的单位：
(i)226981
(ii)13824
(iii)571787
(iv)175616
解决方案：
(i) 226981
Since the unit digit of the given number is 1
So, the unit digit of cube root of 226981 is 1
(ii) 13824
Since the unit digit of the given number is 4
So, the unit digit of cube root of 13824 is 4
(iii) 571787
Since the unit digit of the given number is 7
So, the unit digit of cube root of 571787 is 7
(iv) 175616
Since the unit digit of the given number is 6
So, the unit digit of cube root of 175616 is 6
问题16.在第15题中找到每个数字的立方根的十位数。
(i)226981
(ii)13824
(iii)571787
(iv)175616
解决方案：
(i) 226981
Since the unit digit of the given number is 1
So, the unit digit of cube root of 226981 is 1
Let’s remove the unit, tens, and hundreds digit of the given number, we get 226
As we know the number 226 lies between cube root of 6 and 7 (63 < 226 < 73)
So, 6 is the largest number whose cube root will be less than or equal to 226
Hence, the tens digit of the cube root of 226981 is 6
(ii) 13824
Since the unit digit of the given number is 4
So, the unit digit of cube root of 13824 is 4
Let’s remove the unit, tens, and hundreds digit of the given number, we get 13
As we know the number 13 lies between cube root of 2 and 3 (23 < 13 < 33)
So, 2 is the largest number whose cube root will be less than or equal to 13
Hence, the tens digit of the cube root of 13824 is 2
(iii) 571787
Since the unit digit of the given number is 7
So, the unit digit of cube root of 571787 is 3
Let’s remove the unit, tens, and hundreds digit of the given number, we get 571
As we know the number 571 lies between cube root of 8 and 9 (83 < 571 < 93)
So, 8 is the largest number whose cube root will be less than or equal to 571
Hence, the tens digit of the cube root of 571787 is 8
(iv) 175616
Since the unit digit of the given number is 6
So, the unit digit of cube root of 175616 is 6
Let’s remove the unit, tens, and hundreds digit of the given number, we get 175
As we know the number 175 lies between cube root of 5 and 6 (53 < 571 < 63)
So, 5 is the largest number whose cube root will be less than or equal to 175
Hence, the tens digit of the cube root of 175616 is 5