[3 marks sum]

Question 13 Marks

Find the cube$-$root of $700 \times 2 \times 49 \times 5$.

Answer

$700 \times 2 \times 49 \times 5$

$2$	$700$
$2$	$350$
$5$	$175$
$5$	$35$
$7$	$7$
	$1$

$= 2 \times 2 \times 5 \times 5 \times 7 \times 2 \times 7 \times 7 \times 5$
$= (2 \times 2 \times 2) \times (5 \times 5 \times 5) \times (7 \times 7 \times 7)$
$= 2 \times 5 \times 7$
$= 70$

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Question 23 Marks

Find the cube$-$root of $9.261$

Answer

$9.261$
$ =\sqrt[3]{\frac{9261}{1000}}$
$ =\sqrt{\frac{3 \times 3 \times 3 \times 7 \times 7 \times 7}{10 \times 10 \times 10}}$

$3$	$9261$
$3$	$3087$
$3$	$1029$
$7$	$343$
$7$	$49$
$7$	$7$
	$1$

$ =\frac{3 \times 7}{10}$
$ =\frac{21}{10}$
$ =2.1$

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Question 33 Marks

Find the cube$-$root of $2.744$

Answer

$2.744$
$=\sqrt[3]{\frac{2744}{1000}}$

$2$	$2744$
$2$	$1372$
$2$	$686$
$7$	$343$
$7$	$49$
$7$	$7$
	$1$

$ =\sqrt[3]{\frac{2 \times 2 \times 2 \times 7 \times 7 \times 7}{10 \times 10 \times 10}}$
$ =\frac{2 \times 7}{10}$
$ =\frac{14}{10}$
$ =1.4$

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Question 43 Marks

Find the cube$-$root of $-2744000$

Answer

$ -2744000$
$ =\sqrt[3]{-2744000}$

$2$	$2744000$
$2$	$1372000$
$2$	$686000$
$7$	$343000$
$7$	$49000$
$7$	$7000$
$10$	$1000$
$10$	$100$
$10$	$10$
	$1$

$ =\sqrt{-2 \times-2 \times-2 \times-7 \times-7 \times-7 \times-10 \times-10 \times-10}$
$ =-2 \times-7 \times-10$
$ =-140$

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Question 53 Marks

Find the cube$-$root of $-5832$

Answer

$ -5832$
$ =\sqrt[3]{-5832}$

$2$	$5832$
$2$	$2916$
$2$	$1458$
$3$	$729$
$3$	$243$
$3$	$81$
$3$	$27$
$3$	$9$
$3$	$3$
	$1$

$ =\sqrt{-2 \times-2 \times-2 \times-3 \times-3 \times-3 \times-3 \times-3 \times-3}$
$ =-2 \times-3 \times-3$
$ =-18$

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Question 63 Marks

Find the cube$-$root of $\frac{27}{64}$

Answer

$ \frac{27}{64}$
$ =\sqrt[3]{\frac{27}{64}}$
$ =\frac{\sqrt{3 \times 3 \times 3}}{\sqrt{4 \times 4 \times 4}}$
$ =\frac{3}{4}$

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Question 73 Marks

Find the cube$-$root of $3375$.

Answer

$ 3375$
$ =\sqrt[3]{3375}$
$ =(5 \times 5 \times 5) \times(3 \times 3 \times 3)$
$ =5 \times 3$
$ =15$

$5$	$3375$
$5$	$675$
$5$	$135$
$3$	$27$
$3$	$9$
$3$	$3$
	$1$

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Question 83 Marks

Find the cube$-$root of $8000.$

Answer

$ 8000$
$ =\sqrt[3]{8000}$
$ =(4 \times 4 \times 4) \times(5 \times 5 \times 5)$
$ =4 \times 5$
$ =20$

$4$	$8000$
$4$	$2000$
$4$	$500$
$5$	$125$
$5$	$25$
$5$	$5$
	$1$

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Question 93 Marks

Find the cube$-$root of $9261$.

Answer

$ 9261$
$ =\sqrt[3]{9261}$
$ =(3 \times 3 \times 3) \times(7 \times 7 \times 7)$
$ =3 \times 7$
$ =21$

$3$	$9261$
$3$	$3087$
$3$	$1029$
$7$	$343$
$7$	$49$
$7$	$7$
	$1$

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Question 103 Marks

Find the cube$-$root of $729.$

Answer

$ 729$
$ =\sqrt[3]{729}$
$ =(3 \times 3 \times 3) \times(3 \times 3 \times 3)$
$ =3 \times 3$
$ =9$

$3$	$729$
$3$	$243$
$3$	$81$
$3$	$27$
$3$	$9$
$3$	$3$
	$1$

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Question 113 Marks

Find the smallest number by which $27783$ be multiplied to get a perfect square number.

Answer

$3$	$27783$
$3$	$9261$
$3$	$3087$
$3$	$1029$
$7$	$343$
$7$	$49$
$7$	$7$
	$1$

$= 3 \times 3 \times 3 \times 3 \times 7 \times 7 \times 7$
$= (3 \times 3 \times 3) \times (7 \times 7 \times 7) \times 3$
Clearly, $27783$ must be multiplied by $3 \times 3$
$= 9$

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Question 123 Marks

Find the smallest number by which $8768$ must be divided so that the quotient is a perfect cube.

Answer

The prime factor of $8768$ are

$2$	$8768$
$2$	$4384$
$2$	$2192$
$2$	$1096$
$2$	$548$
$2$	$274$
$137$	$137$
	$1$

$= 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 137$
$= (2 \times 2 \times 2) \times (2 \times 2 \times 2) \times 137$
Clearly, $8768$ must be divided by $137$.

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Question 133 Marks

Find the cube of: $2 \frac{1}{2}$

Answer

$ 2 \frac{1}{2}$
$ =\left(2 \frac{1}{2}\right)^3$
$ =\left(\frac{5}{2}\right)^3$
$ =\frac{5 \times 5 \times 5}{2 \times 2 \times 2}$
$ =\frac{125}{8}$
$ =15 \frac{5}{8}$

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Question 143 Marks

Find the cube of: $1 \frac{2}{7}$

Answer

$ 1 \frac{2}{7}$
$ =\left(1 \frac{2}{7}\right)^3$
$ =\left(\frac{1 \times 7+2}{7}\right)^3$
$ =\left(\frac{9}{7}\right)^3$
$ =\frac{9 \times 9 \times 9}{7 \times 7 \times 7}$
$=\frac{729}{343}$
$=2 \frac{43}{343}$

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Question 153 Marks

Find the cube of: $ 0.8$

Answer

$ 0.8$
$ =(0.8)^3$
$ =\left(\frac{8}{10}\right)^3$
$ =\frac{8 \times 8 \times 8}{10 \times 10 \times 10}$
$ =\frac{512}{1000}$
$ =0.512$

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Question 163 Marks

Find the cube of: $0.02$

Answer

$ 0.02$
$ =(0.02)^3$
$ =\left(\frac{2}{10}\right)^3$
$ =\frac{2 \times 2 \times 2}{100 \times 100 \times 100}$
$ =\frac{8}{1000000}$
$ =0.000008$

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Question 173 Marks

Find the cube of: $ 0.12$

Answer

$ 0.12$
$ =(0.12)^3$
$ =\left(\frac{12}{10}\right)^3$
$ =\frac{12 \times 12 \times 12}{100 \times 100 \times 100}$
$ =\frac{1728}{1000000}$
$ =0.001728$

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Question 183 Marks

Find the cube of: $ 2.5$

Answer

$ 2.5$
$ =(2.5)^3$
$ =\left(\frac{25}{10}\right)^3$
$ =\frac{25 \times 25 \times 25}{10 \times 10 \times 10}$
$ =\frac{15625}{1000}$
$ =15.625$

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Question 193 Marks

Find the cube of: $ 0.4$

Answer

$ 0.4$
$ =(0.4)^3$
$ =\left(\frac{4}{10}\right)^3$
$ =\frac{4 \times 4 \times 4}{10 \times 10 \times 10}$
$ =\frac{64}{1000}$
$ =0.064$

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Question 203 Marks

Find the cube of: $ 2.1$

Answer

$2.1$
$=(2.1)^3$
$=\left(\frac{21}{10}\right)^3$
$=\frac{21 \times 21 \times 21}{10 \times 10 \times 10}$
$=\frac{9261}{1000}$
$=9.261$

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Question 213 Marks

Find if the following number is a perfect cube? $1728$

Answer

$1728$

$2$	$1728$
$2$	$864$
$2$	$432$
$2$	$216$
$2$	$108$
$2$	$54$
$3$	$27$
$3$	$9$
$3$	$3$
	$1$

$\because 1728 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3$
$= (2)^3 \times (2)^3 \times (3)^3$
$\because 1728$ is a perfect cube.

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Question 223 Marks

Find if the following number is a perfect cube? $24000$

Answer

$24000$
$\because 24000 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 5 \times 5 \times 5$

$2$	$24000$
$2$	$12000$
$2$	$6000$
$2$	$3000$
$2$	$1500$
$2$	$750$
$3$	$375$
$5$	$125$
$5$	$25$
$5$	$5$
	$1$

$= (2)^2 \times (2)^3 \times (5)^3 \times 3$
$\therefore 24000$ is not a perfect cube.

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Question 233 Marks

Find if the following number is a perfect cube? $1331$

Answer

$1331$

$11$	$1331$
$11$	$121$
$11$	$11$
	$1$

$\therefore 1331 = 11 \times 11 \times 11 = (11)^3$
$\therefore 1331$ is a perfect cube.

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Question 243 Marks

Find if the following number is a perfect cube? $243$

Answer

$243$

$3$	$243$
$3$	$81$
$3$	$27$
$3$	$9$
$3$	$3$
	$1$

$\because 243 = 3 \times 3 \times 3 \times 3 \times 3$
$= (3 \times 3 \times 3) \times 3 \times 3$
$= 3^3 \times 3 \times 3$
$\therefore 279$ is not a perfect cube.

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Question 253 Marks

Which is the smallest number that must be multiplied to $77175$ to make it a perfect cube?

Answer

The prime factor of $77175$ are

$3$	$77175$
$3$	$25725$
$5$	$8575$
$5$	$1715$
$7$	$343$
$7$	$49$
$7$	$7$
	$1$

$= 3 \times3 \times5 \times5 \times7 \times7 \times7$
$= (7 \times7 \times7) \times3 \times3 \times5 \times5$
Clearly, $77175$ must be multiplied by $3 \times5$
$= 15$

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Question 263 Marks

With what least number must $8640$ be divided so that the quotient is a perfect cube?

Answer

The prime factors of $8640$ are

$2$	$8640$
$2$	$4320$
$2$	$2160$
$2$	$1080$
$2$	$540$
$2$	$270$
$3$	$135$
$3$	$45$
$3$	$15$
$5$	$5$
	$1$

$= 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 5$
$= (2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (3 \times 3 \times 3) \times 5$
Clearly, $8640$ must be divided by $5.$ So, that the quotient is a perfect cube.

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