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MATHS [4 marks sum]

Cubes and Cube Roots · ICSE STD 8 (ICSE - English Medium). 10 questions.

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[4 marks sum]

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Question 14 Marks
Find the cube$-$root of $-175616$
Answer
$-175616$
$2$ $175616$
$2$ $27808$
$2$ $43904$
$2$ $21952$
$2$ $10976$
$2$ $5488$
$2$ $2744$
$2$ $1372$
$2$ $686$
$7$ $343$
$7$ $49$
$7$ $7$

 		 $1$
$= -[(2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (7 \times 7 \times 7)]$
$=-[2 \times 2 \times 2 \times 7]$
$= -56$
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Question 24 Marks
Find the cube$-$root of $250.047$
Answer
$250.047$
$=\frac{250047}{1000}$
$3$ $250047$
$3$ $83349$
$3$ $27783$
$3$ $9261$
$3$ $3087$
$3$ $1029$
$7$ $343$
$7$ $49$
$7$ $7$

 		 $1$
$ =\frac{(3 \times 3 \times 3) \times(3 \times 3 \times 3) \times(7 \times 7 \times 7)}{(10 \times 10 \times 10)}$
$ =\frac{3 \times 3 \times 7}{10}$
$ =\frac{63}{10}$
$ =6.3$
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Question 34 Marks
Find the cube$-$root of $-216 \times 1728$
Answer
-$216 \times 1728$
$2$ $216$
$2$ $108$
$2$ $54$
$3$ $27$
$3$ $9$
$3$ $3$

 		 $1$
$2$ $1728$
$2$ $864$
$2$ $432$

 		 $216$
$= -(2 \times 2 \times 2 \times 3 \times 3 \times 3) \times (2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3)$
$= -2 \times 3 \times 2 \times 2 \times 3$
$= -72$
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Question 44 Marks
What is the least number by which $30375$ should be multiplied to get a perfect cube? 
Answer
The prime factors of $30375$ are
$3$ $30375$
$3$ $10125$
$3$ $3375$
$3$ $1125$
$3$ $375$
$5$ $125$
$5$ $25$
$5$ $5$

 		 $1$
$= 3 \times 3 \times 3 \times 3 \times 3 \times 5 \times 5 \times 5$
$= (3 \times 3 \times 3) \times (5 \times 5 \times 5) \times 3 \times 3$
Clearly, $30375$ must be multiplied with $3$
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Question 54 Marks
Find the smallest number by which $26244$ may be divided so that the quotient is a perfect cube.
Answer
The prime factors of $26244$ are
$2$ $26244$
$2$ $13122$
$3$ $6561$
$3$ $2187$
$3$ $729$
$3$ $243$
$3$ $81$
$3$ $27$
$3$ $9$
$3$ $3$

 		 $1$
$= 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3$
$= (3 \times 3 \times 3) \times (3 \times 3 \times 3) \times 3 \times 3 \times 2 \times 2$
Clearly, $26244$ must be divided by
$3 \times 3 \times 2 \times 2 = 36$
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Question 64 Marks
Find the cube$-$root of $- 15.625.$
Answer
$ -15.625$
$ =\sqrt[3]{\frac{-15625}{1000}}$
$ =\sqrt{\frac{-(5 \times 5 \times 5) \times(5 \times 5 \times 5)}{10 \times 10 \times 10}}$
$ =\frac{-5 \times 5}{10}$
$ =\frac{-25}{10}$
$ =-2.5$
$5$ $15625$
$5$ $3125$
$5$ $625$
$5$ $125$
$5$ $25$
$5$ $5$
  $1$
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Question 74 Marks
Find the cube$-$root of $4096.$
Answer
$ 4096$
$ =\sqrt[3]{4096}$
$ =(2 \times 2 \times 2) \times(2 \times 2 \times 2) \times(2 \times 2 \times 2) \times(2 \times 2 \times 2)$
$ =2 \times 2 \times 2 \times 2$
$ =16$
$2$ $4096$
$2$ $2048$
$2$ $1024$
$2$ $512$
$2$ $256$
$2$ $128$
$2$ $64$
$2$ $32$
$2$ $16$
$2$ $8$
$2$ $4$
$2$ $2$

 		 $1$
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Question 84 Marks
Find the cube$-$root of $1728.$
Answer
$ 1728$
$ =\sqrt[3]{1728}$
$ =(2 \times 2 \times 2) \times(2 \times 2 \times 2) \times(3 \times 3 \times 3)$
$ =2 \times 2 \times 3$
$ =12$
$2$ $1728$
$2$ $864$
$2$ $432$
$2$ $216$
$2$ $108$
$2$ $54$
$3$ $27$
$3$ $9$
$3$ $3$

 		 $1$
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Question 94 Marks
Which of the following are cubes of an odd number. $216, 729, 3375, 8000, 125, 343, 4096$ and $9261$.
Answer
$\because 216 = 2 \times 2 \times 2 \times 3 \times 3 \times 3$
$2$ $216$
$2$ $108$
$2$ $54$
$3$ $27$
$3$ $9$
$3$ $3$
  $1$
$= (2)^3 \times (3)^3$
$= (6)^3$
$\because 729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3$
$3$ $729$
$3$ $243$
$3$ $81$
$3$ $27$
$3$ $9$
$3$ $3$
  $1$
$= (3)^3 \times (3)^3$
$= (9)^3$
$\because 3375 = 5 \times 5 \times 5 \times 3 \times 3 \times 3$​​​​​​​
$5$ $3375$
$5$ $675$
$5$ $135$
$3$ $27$
$3$ $9$
$3$ $3$
  $1$
$= (5)^3 \times (3)^3$
$= (15)^3$
$\because 8000 = 20 \times 20 \times 20$
$= (20)^3$​​​​​​​​​​​​​​
$5$ $125$
$5$ $25$
$5$ $5$
  $1$
$125 = 5 \times 5 \times 5$
$= (5)^3$
$\because 343 = 7 \times 7 \times 7$
$= (7)^3$​​​​​​​​​​​​​​​​​​​​​
$7$ $343$
$7$ $49$
$7$ $7$
  $1$
$\because 4096 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$​​​​​​​
$2$ $4096$
$2$ $2048$
$2$ $1024$
$2$ $512$
$2$ $256$
$2$ $128$
$2$ $64$
$2$ $32$
$2$ $16$
$2$ $8$
$2$ $4$
$2$ $2$
  $1$
$= (2)^3 \times (2)^3 \times (2)^3 \times (2)^3$
$=(16)^3$​​​​​​​
Cubes of an odd number are $729, 3375, 125, 343, 9261$.
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Question 104 Marks
Which of the following are cubes of an even number $216, 729, 3375, 8000, 125, 343, 4096$ and $9261$.
Answer
$\because 216 = 2 \times 2 \times 2 \times 3 \times 3 \times 3$
$2$ $216$
$2$ $108$
$2$ $54$
$3$ $27$
$3$ $9$
$3$ $3$
  $1$
$= (2)^3 \times (3)^3$
$= (6)^3$
$\because 729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3$
$3$ $729$
$3$ $243$
$3$ $81$
$3$ $27$
$3$ $9$
$3$ $3$
  $1$
$= (3)^3 \times (3)^3$
$= (9)^3$
$\because 3375 = 5 \times 5 \times 5 \times 3 \times 3 \times 3$
$5$ $3375$
$5$ $675$
$5$ $135$
$3$ $27$
$3$ $9$
$3$ $3$
  $1$
$= (5)^3 \times (3)^3$
$= (15)^3$
$\because 8000 = 20 \times 20 \times 20$
$= (20)^3$
$5$ $125$
$5$ $25$
$5$ $5$
  $1$
$125 = 5 \times 5 \times 5$
$= (5)^3$
$\because 343 = 7 \times 7 \times 7$
$= (7)^3$
$7$ $343$
$7$ $49$
$7$ $7$
  $1$
$\because 4096 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$
$2$ $4096$
$2$ $2048$
$2$ $1024$
$2$ $512$
$2$ $256$
$2$ $128$
$2$ $64$
$2$ $32$
$2$ $16$
$2$ $8$
$2$ $4$
$2$ $2$
  $1$
$= (2)^3 \times (2)^3 \times (2)^3 \times (2)^3$
$=(16)^3$
Cubes of an even number are $216, 8000, 4096$.
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