# Cubes and Cube Roots - Class 8 - Mathematics

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## Examples - Cubes and Cube Roots | NCERT | Mathematics | Class 8

Is 243 a perfect cube?

No, $243$ is not a perfect cube. The nearest cubic numbers to $243$ are $216$ and $343$.

Here is a plot of the cubic numbers, showing how $243$ is not among them:

And here is a number line indicating the position of $243$ relative to nearby cubic numbers:

Is 392 a perfect cube? If not, find the smallest natural number by which 392 must be multiplied so that the product is a perfect cube.

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Sign up nowIs 53240 a perfect cube? If not, then by which smallest natural number should 53240 be divided so that the quotient is a perfect cube?

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Sign up nowIs 1188 a perfect cube? If not, by which smallest natural number should 1188 be divided so that the quotient is a perfect cube?

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Sign up nowIs 68600 a perfect cube? If not, find the smallest number by which 68600 must be multiplied to get a perfect cube.

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Find the cube root of 8000.

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Find the cube root of 13824 by prime factorisation method.

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## Exercise 6.1 - Cubes and Cube Roots | NCERT | Mathematics | Class 8

Which of the following numbers are not perfect cubes?

(i) 216

(ii) 128

(iii) 1000

(iv) 100

(v) 46656

The numbers that are **not** perfect cubes among the given options are:

- (ii) $128$
- (iv) $100$

The numbers that **are** perfect cubes are:

- (i) $216$ (it is the cube of $6$)
- (iii) $1000$ (it is the cube of $10$)
- (v) $46656$ (it is the cube of $36$)

Find the smallest number by which each of the following numbers must be multiplied to obtain a perfect cube.

(i) 243

(ii) 256

(iii) 72

(iv) 675

(v) 100

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Find the smallest number by which each of the following numbers must be divided to obtain a perfect cube.

(i) 81

(ii) 128

(iii) 135

(iv) 192

(v) 704

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Parikshit makes a cuboid of plasticine of sides $5 \mathrm{~cm}, 2 \mathrm{~cm}, 5 \mathrm{~cm}$. How many such cuboids will he need to form a cube?

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## Exercise 6.2 - Cubes and Cube Roots | NCERT | Mathematics | Class 8

Find the cube root of each of the following numbers by prime factorisation method.

(i) 64

(ii) 512

(iii) 10648

(iv) 27000

(v) 15625

(vi) 13824

(ix) 175616

(x) 91125

(vii) 110592

(viii) 46656

State true or false.

(i) Cube of any odd number is even.

(ii) A perfect cube does not end with two zeros.

(iii) If square of a number ends with 5, then its cube ends with 25 .

(iv) There is no perfect cube which ends with 8 .

(v) The cube of a two digit number may be a three digit number.

(vi) The cube of a two digit number may have seven or more digits.

(vii) The cube of a single digit number may be a single digit number.

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## Extra Questions - Cubes and Cube Roots | NCERT | Mathematics | Class 8

Find the cube root of:

(i) $343$

(ii) $1000$

(iii) $2744$

(iv) $74088$

**Answer:**

(i) To find the cube root of $343$, consider prime factors or perfect cubes known: $$ 343 = 7 \times 7 \times 7 = 7^3 $$ Therefore, the cube root is: $$ \sqrt[3]{343} = 7 $$

(ii) For $1000$, since it is a common cubic number, $$ 1000 = 10 \times 10 \times 10 = 10^3 $$ Thus, the cube root is: $$ \sqrt[3]{1000} = 10 $$

(iii) For $2744$, considering known cubes: $$ 2744 = 14 \times 14 \times 14 = 14^3 $$ Therefore, the cube root is: $$ \sqrt[3]{2744} = 14 $$

(iv) For $74088$, breaking down into its factors and recognizing perfect cubes: Hence, its cube root is: $$ \sqrt[3]{74088} = 2 \times 23 = 42 $$

Thus, the cube roots are:

**7****10****14****42**

Using prime factorization, find which of the following are perfect cubes.

a) 128

b) 343

c) 729

d) 1331

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"There are 8 pipes which take 26 hours to fill a big drum. But when the operations were supposed to begin, two pipes were damaged. It so happened that only one could be retrieved back to a usable state. What will be the time taken to fill the drum with the existing pipes?"

A) 29.7 hours

B) 30.7 hours

C) 28.7 hours

D) 27.7 hours.

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Which of the following is an example of a non-polyhedron?

A. Cube

B. Cone

C. Triangular prism

D. Triangular pyramid

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The arrangement $A$ $B$ $C$ $\ldots$ $A$ $B$ $C$ $\ldots$ $A$ $B$ $C$ $\ldots$ is referred to as

A) octahedral close packing

B) hexagonal close packing

C) tetragonal close packing

D) cubic close packing

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2000 cubic centimeters $=$ $\qquad$ cubic meters.

(A) 0.2

(B) 0.02

(C) 0.002

(D) 0.0002

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Simplify: $\frac{\sqrt[4024]{44}}{\sqrt[4]{4}}$

A) 4

B) 2

C) 16

D) 64

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Find the value of $\left(36^{3}\right)^{-3}$.

A) $\left(36^{-9}\right)$

B) $\left(36^{9}\right)$

C) $\left(6^{18}\right)$

D) $\left(6^{-18}\right)$

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$\tan \frac{\pi}{3} = \sqrt{3}$

A) True

B) False

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$(a + b)$ and $(a - b)$ are two diagonals of a rhombus; $a, b > 0$. What is the value of $a^{3} + b^{3}$, if the area of the rhombus is $30 , \mathrm{cm}^{2}$? Given that one of the diagonals is $6 , \mathrm{cm}$.

A) $60 , \mathrm{cm}$

B) $520 , \mathrm{cm}$

C) $504 , \mathrm{cm}$

D) $80 , \mathrm{cm}$

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Two towers $A$ and $B$ are standing at some distance apart. From the top of tower $A$, the angle of depression of the foot of tower $B$ is found to be $30^\circ$. From the top of tower $B$, the angle of depression of the foot of tower $A$ is found to be $60^\circ$. If the height of tower $B$ is '$h$' meters, then the height of tower $A$ in terms of '$h$' is

(A) $2h$ meters (B) $\frac{h}{3}$ meters (C) $\sqrt{3} h$ meters (D) $\frac{h}{\sqrt{3}}$ meters

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The value of $\cos^{3}\left(60^{\circ}-A\right)-\cos^{3}\left(60^{\circ}+A\right)$ is

(A) $3\sqrt{3}$

(B) $\frac{3\sqrt{3}}{4} \sin A$

(C) $\frac{3\sqrt{3}}{4}$

(D) $3\sqrt{3} \sin A$