Explore topic-wise InterviewSolutions in Current Affairs.

To evaluate the cube of (3.5)³

We need to multiply the given number three times i.e. 3.5 × 3.5 × 3.5 = 42.875

We now have to convert the given number into fraction we get,

(42875/1000) = 343/8

∴ The cube of 3.5 is 42.875

(i) \(-11\)

 = \((-11)^3 = -11\times-11\times-11=-1331\)

(ii) \(-12\)

 = \((-2)^3 =-12\times-12\times-12=-1728\)

(iii) \(-21\)

 = \((-21)^3 = -12\times-21\times-21=-9261\)

According to the question, 

Area of square = Area of rectangle

side² = 240 x 70

side = \(\sqrt{240\times70}\)

 = \(\sqrt{10\times10\times2\times2\times2\times3\times7}\)

 = \(20\sqrt{42}\)

= 20 x 6.48

= 129.69 m

To evaluate the cube of (1.2)³

We need to multiply the given number three times i.e. 1.2 × 1.2 × 1.2 = 1.728

We now have to convert the given number into fraction we get,

(1728/1000) = 216/125

∴ The cube of 1.2 is 1.728

From above pattern, we sec cube of 1 consists is 1 odd number

Cube of 2 Consists 2 odd numbers 3, 5

Cube of 3 Consists 3 odd numbers 7, 9, 11 

Cube of 4 Consists 4 odd numbers 13, 15, 17, 19 

_____________________________ 

∴ Cube of 10 consists 10 consecutive odd numbers.

To evaluate the cube of (8)³

We need to multiply the given number three times i.e. 8 × 8 × 8 = 512

∴ The cube of 8 is 512

(i) 216 

= 2³ × 3³ 

= 6³

It’s a cube of even natural number.

(ii) 512 

= 2⁹ 

= (2³)³ 

= 8³

It’s a cube of even natural number.

(iii) 729 

= 3³ × 3³ 

= 9³

It’s not a cube of even natural number.

(iv) 1000 

= 10³

It’s a cube of even natural number.

(v) 3375 

= 3³ × 5³ 

= 15³

It’s not a cube of even natural number.

(vi) 13824 

= 2⁹ × 3³

= (2³)³ × 3³

= 8³ × 3³ 

= 24³

It’s a cube of even natural number.

(i) 125 = 5 × 5 × 5 × 5 = 5³

It’s a cube of odd natural number. 

(ii) 343 = 7 × 7 × 7 = 7³ 

It’s a cube of odd natural number. 

(iii) 1728 = 2⁶ × 3³ = 4³ × 3³ = 12³ 

It’s not a cube of odd natural number. As 12 is even number. 

(iv) 4096 = 2¹² = (2⁶)² = 64² 

Its not even a cube. (v) 32768 = 2¹⁵ = (2⁵)³ = 32³ 

It’s a cube of odd natural number. As 32 is an even number. 

(vi) 6859 = 19 × 19 × 19 = 19³ 

It’s a cube of odd natural number.

(i) 7

Cube of 7 = 7 × 7 × 7 = 343 

(ii) 12 

Cube of 12 = 12 × 12 × 12 = 1728 

(iii) 16 

Cube of 16 = 16 × 16 ×16 = 4096 

(iv) 21 

cube of 21 = 21 × 21 × 21 = 9261 

(v) 40 

Cube of 40 = 40 × 40 × 40 = 64000 

(vi) 55 

Cube of 55 = 55 × 55 × 55 = 166375 

(vii) 100 

Cube of 100 = 100 × 100 × 100 = 1000000 

(viii) 302

To find cube of 302 we make it in form (a + b)³, Which make caluclation easier

 = (a + b) = a³ + b³ + 3a²b + 3ab² = (300 + 2)³ + 3x300² x 2 + 3x300x²

 = 27000000 + 8 + 540000 + 3600 = 27362408.

(ix) 301

To find cube of 301 we make it in form (a + b)³, Which make calculation easier.

 = (a + b)³ = a³ + b³ + 3a²b + 3ab² = (300 + 1)³ = 300³ + 1³ + 3x300²x1 + 3x300x1³

= 27000000 + 1 + 270000 + 900 = 27180601.

To evaluate the cube of (15)³

We need to multiply the given number three times i.e. 15 × 15 × 15 = 3375

∴ The cube of 15 is 3375

To evaluate the cube of (60)³

We need to multiply the given number three times i.e. 60 × 60 × 60 = 216000

∴ The cube of 60 is 216000

According to given pattern,

 = 1³ + 2³ + 3² + ......+n³) = (1 + 2 + 3 + ......+ n)²

Here n = 10, so,

 = (1³ + 2³ + 3² + .....+ 9³ + 10³) = (1 + 2 + 3 +.......+9 + 10)²

  = (1³ + 2³ + 3³ +......+ 9³ + 10³) = (55²) = 55x55 = 3025.

(i) 250

250 = 25×100

By using cube root table 250 would be in column ∛10x against 25.

We get,

∛250 = 6.3

∴ the answer is 6.3

(ii) 5112

∛5112 = ∛2×2×2×3×3×71

= ∛2³×3²×71

= 2 × ∛3² × ∛71

= 2 × ∛9 × ∛71

From cube root table we get,

∛9 = 2.080

∛71 = 4.141

∛5112 = 2 × ∛9 × ∛71

= 2 × 2.080 × 4.141

= 17.227

∴ the answer is 17.227

(iii) 9800

∛9800 = ∛98 × ∛100

From cube root table we get,

∛98 = 4.610

∛100 = 4.642

∛9800 = ∛98 × ∛100

= 4.610 × 4.642

= 21.40

∴ the answer is 21.40

(iv) 732

∛732 Now,

We know that value of ∛732 will lie between ∛730 and ∛740

From cube root table we get,

∛730 = 9.004

∛740 = 9.045

By using unitary method,

Difference between the values (740 – 730 = 10)

So, the difference in cube root values will be = 9.045 – 9.004 = 0.041

Difference between the values (732 – 730 = 2)

So, the difference in cube root values will be = (0.041/10) ×2 = 0.008

∛732 = 9.004+0.008 = 9.012

∴ the answer is 9.012

(v) 7342

∛7342 Now,

We know that value of ∛7342 will lie between ∛7300 and ∛7400

From cube root table we get,

∛7300 = 19.39

∛7400 = 19.48

By using unitary method,

Difference between the values (7400 – 7300 = 100)

So, the difference in cube root values will be = 19.48 – 19.39 = 0.09

Difference between the values (7342 – 7300 = 42)

So, the difference in cube root values will be ,

= (0.09/100) × 42 

= 0.037

∛7342 = 19.39+0.037 

= 19.427

∴ the answer is 19.427

(i) 125

 = 5 × 5 × 5 × 5 

= 5³

It’s a cube of odd natural number.

(ii) 343 

= 7 × 7 × 7 

= 7³

It’s a cube of odd natural number.

(iii) 1728 

= 2⁶ × 3³

= 4³ × 3³ 

= 12³

It’s not a cube of odd natural number. As 12 is even number.

(iv) 4096 

= 2¹² 

= (2⁶)² 

= 64²

Its not a cube of odd natural number. As 64 is an even number.

(v) 32768 

= 2¹⁵ 

= (2⁵)³ 

= 32³

It’s not a cube of odd natural number. As 32 is an even number.

(vi) 6859 

= 19 × 19 × 19 

= 19³

It’s a cube of odd natural number.

Cube of natural numbers upto 10 are as follows.

1³ = 1 × 1 × 1 = 1

2³ = 2 × 2 × 2 = 8

3³ = 3 × 3 × 3 = 27

4³ = 4 × 4 × 4 = 64

5³ = 5 × 5 × 5 = 125

6³ = 6 × 6 × 6 = 216

7³ = 7 × 7 × 7 = 343

8³ = 8 × 8 × 8 = 512

9³ = 9 × 9 × 9 = 729

10³ = 10 × 10 × 10 = 1000

From above results we can see that, 

(i) Cubes of all odd natural numbers are odd. 

(ii) Cubes of all even natural numbers are even.

To evaluate the cube of (21)³

We need to multiply the given number three times i.e. 21 × 21 × 21 = 9261

∴ The cube of 21 is 9261

(i) 1100 = 11×100

∛1100 = ∛(11×100) 

= ∛11 × ∛100

By using cube root table,

We get,

∛11 = 2.224

∛100 = 4.6642

∛1100 = ∛11 × ∛100

= 2.224 × 4.642

= 10.323

∴ the answer is 10.323

(ii) 780

780 = 78×10

By using cube root table 780 would be in column ∛10x against 78.

We get,

∛780 = 9.205

(iii) 7800

7800 = 78×100

∛7800 = ∛(78×100) 

= ∛78 × ∛100

By using cube root table,

We get,

∛78 = 4.273

∛100 = 4.6642

∛7800 = ∛78 × ∛100

= 4.273 × 4.642

= 19.835

∴ the answer is 19.835

(iv) 1346

1346 = 2×673

∛1346 = ∛(2×676)

 = ∛2 × ∛673

Since, 670<673<680 = ∛670 < ∛673 < ∛680

By using cube root table,

∛670 = 8.750

∛680 = 8.794

For the difference (680-670) which is 10.

So the difference in the values = 8.794 – 8.750 = 0.044

For the difference (673-670) which is 3.

So the difference in the values = (0.044/10) × 3 = 0.0132

∛673 = 8.750 + 0.013 = 8.763

∛1346 = ∛2 × ∛673

= 1.260 × 8.763

= 11.041

∴ the answer is 11.041

(i) To calculate the cube of (1.2)³ 

We have to multiply the given number three times; 

= (1.2 × 1.2 × 1.2) = 1.728 

Now by converting it into fraction we get,

= \(\frac{1728}{1000}\)

= \(\frac{216}{125}\)

(ii) To calculate the cube of (3.5)³ 

We have to multiply the given three times; 

= (3.5 × 3.5 × 3.5) = 42.875 

Now by converting it into fraction we get,

= \(\frac{42875}{1000}\)

= \(\frac{343}{8}\)

(iii) To calculate the cube of (0.8)³ 

We have to multiply the given number by its power; 

= (0.8 × 0.8 × 0.8) = 0.512 

Now by converting it into fraction we get,

= \(\frac{512}{1000}\)

= \(\frac{64}{125}\)

(iv) To calculate the cube of (0.05)³ 

We have to multiply the given number by its power; 

= (0.05 × 0.05 × 0.05) = 0.000125 

Now by converting it into fraction we get,

= \(\frac{125}{1000000}\)

= \(\frac{1}{8000}\)

(i) 7

As we know that 7 lies between 1 and 100 so by using cube root table we get,

∛7 = 1.913

∴ the answer is 1.913

(ii)  70

As we know that 70 lies between 1 and 100 so by using cube root table from column x

We get,

∛70 = 4.121

∴ the answer is 4.121

(iii) 700

700 = 70×10

By using cube root table 700 will be in the column ∛10x against 70.

So we get,

∛700 = 8.879

∴ the answer is 8.879

(iv) 7000

7000 = 70×100

∛7000 = ∛(7×1000) 

= ∛7 × ∛1000

By using cube root table,

We get,

∛7 = 1.913

∛1000 = 10

∛7000 = ∛7 × ∛1000

= 1.913 × 10

= 19.13

∴ the answer is 19.13

(i) To calculate the cube of (8)³ 

We have to multiply the given number three times; 

= (8 × 8 × 8) = 512 

So, 512 is the cube of 8. 

(ii) (15)³ 

First multiply the given number three times; 

= (15 × 15 ×15) = 3375 

So, 3375 is the cube of 15 

(iii) (21)³ 

First multiply the given number three times; 

= (21 × 21 × 21) = 9261 

9261 is the cube of 21. 

(iv) (60)³ 

First multiply the given number three times; 

= (60 × 60 × 60) = 216000 

216000 is the cube of 60.

(i) 7

 = 7× 7 × 7 

= 343

(ii) 12

 = 12× 12× 12 

= 1728

(iii) 16

= 16× 16× 16 

= 4096

(iv) 21

Cube of 21 is

= 21 × 21 × 21 

= 9261

(v) 40

= 40× 40× 40 

= 64000

(vi) 55

Cube of 55 is

= 55× 55× 55

= 166375

(vii) 100

Cube of 100 is

= 100× 100× 100 

= 1000000

(viii) 302

Cube of 302 is

= 302× 302× 302 

= 27543608

(ix) 301

Cube of 301 is

 = 301× 301× 301 

= 27270901

As we know that the first 5 natural numbers which are multiple of 3 are 3, 6, 9, 12 and 15

So now,

The cube of 3, 6, 9, 12 and 15

3³ = 3 × 3 × 3 

= 27

6³ = 6 × 6 × 6 

= 216

9³ = 9 × 9 × 9 

= 729

12³ = 12 × 12 × 12

= 1728

15³ = 15 × 15 × 15 

= 3375

All the cubes are divisible by 27

∴ “The cube of a natural number which is a multiple of 3 is a multiple of 27’

1³ = 1 × 1 × 1 

= 1

2³ = 2 × 2 × 2 

= 8

3³ = 3 × 3 × 3 

= 27

4³ = 4 × 4 × 4 

= 64

5³ = 5 × 5 × 5 

= 125

6³ = 6 × 6 × 6 

= 216

7³ = 7 × 7 × 7

= 343

8³ = 8 × 8 × 8 

= 512

9³ = 9 × 9 × 9 

= 729

10³ = 10 × 10 × 10

 = 1000

Hence , it is clear that ,

(i) Cubes of all odd natural numbers are odd.

(ii) Cubes of all even natural numbers are even.

To evaluate the cube of (0.05)³

We need to multiply the given number three times i.e. 0.05 × 0.05 × 0.05 = 0.000125

We now have to convert the given number into fraction we get,

(125/1000000) = 1/8000

∴ The cube of 1.2 is 0.000125

(i) \((\frac{4}{7})^3\)

By multiplying we get,

= \((\frac{4}{7}\times\frac{4}{7}\times\frac{4}{7})\) 

= \((\frac{64}{343})\)

So, cube of \(\frac{4}{7}\) is \(\frac{64}{343}\)

(ii) \((\frac{10}{11})^3\)

Multiplying the given number three times we get,

= \((\frac{10}{11}\times\frac{10}{11}\times\frac{10}{11})\)

= \((\frac{1000}{1331})\)

(iii) \((\frac{1}{15})^3\)

Multiplying the given number three times we get,

= \((\frac{1}{15}\times\frac{1}{15}\times\frac{1}{15})\)

= \((\frac{1}{3375})\)

(iv) \((1\frac{3}{10})^3\)

Multiplying the given number three times we get,

= \((\frac{13}{10})^3\)

= \((\frac{13}{10}\times\frac{13}{10}\times\frac{13}{10})\)

= \((\frac{2197}{1000})\)

To evaluate the cube of (4/7)³

We need to multiply the given number three times i.e. (4/7) × (4/7) × (4/7) = (64/343)

∴ The cube of (4/7) is (64/343)

13=1 

23 = 8 

33 = 27

 4³= 64 

5³ = 125 

6³ = 216 

7³ = 343 

8³ = 512 

9³ = 729 

10³ = 1000 

∴ It is not possible to say whether the number is a perfect cube (or) not looking at the digit in unit place.

The prime factors of 9261 = 3 × 3 × 3 × 7 × 7 × 7

Now by grouping into three we get, (3 × 3 × 3) × (7 × 7 × 7)

∴ \(\sqrt[3]{((3)^3 × (7)^3)}\) 

= (3 × 7) 

= 21

To evaluate the cube of (10/11)³

We need to multiply the given number three times i.e. (10/11) × (10/11) × (10/11) = (1000/1331)

∴ The cube of (10/11) is (1000/1331)

A perfect cube can be expressed as a product of three numbers of equal factors

Now by resolving the given number into prime factors we get,

256 = 2 × 2 × 2 × 2 × 2× 2 × 2 × 2

Since the given number has more than three factors

∴ 256 is not a perfect cube.

The prime factors of 1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3

Now by grouping into three we get, (2 × 2 × 2) × (2 × 2 × 2) × (3 × 3 × 3)

∴ \(\sqrt[3]{((2)^3 × (2)^3× (3)^3) }\)

= (2 × 2 × 3) 

= 12

A perfect cube can be expressed as a product of three numbers of equal factors

Now by resolving the given number into prime factors we get,

343 = 7 × 7 × 7

Its cube can be expressed as 7³

∴ 343 is a perfect cube.

To evaluate the cube of (1/15)³

We need to multiply the given number three times i.e. (1/15) × (1/15) × (1/15) = (1/3375)

∴ The cube of (1/15) is (1000/3375)

The prime factors of 729 = 3 × 3 × 3 × 3 × 3 × 3

Now by grouping into three we get, (3 × 3 × 3) × (3 × 3 × 3)

∴\(\sqrt[3]{((3)^3 × (3)^3) }\)

= (3 × 3) 

= 9

To evaluate the cube of (1 3/10)³

Firstly we need to convert into proper fraction i.e. (13/10)³

We need to multiply the given number three times i.e. (13/10) × (13/10) × (13/10) = (2197/1000)

∴ The cube of (1 3/10) is (2197/1000)

(i) We have, 

Cube root of \(-125 = \sqrt[3]{-125}\)

\( = -\sqrt[3]{125}\)

\(= -\sqrt[3]{125}\)

\(= - \sqrt[3]{5\times5\times5}\)

\(= -5\)

(ii) We have,

Cube root of -5832

\(= \sqrt[3]{-5832}\)

\(= - \sqrt[3]{5832}\)

So to find out the cube root of 5832, we will use the mehod of unit digits. 

Let’s take number 5832. Unit digit = 2 

So unit digit in the cube root of 5832 = 8 

After striking out the units, tens and hundreds digits of 5832, 

Now we left with 5 only. 

As we know that 1 is the Largest number whose cube is less than or equals to 5. 

So, The tens digit of the cube root of 5832 is 1.

\(\sqrt[3]{5832} = 18\)

\(\sqrt[3]{-5832}\)

\(= -\sqrt[3]{5832}\)

\(= - 18\)

(iii) We have,

\(\sqrt[3]{-2744000}\)

\(= - \sqrt[3]{2744000}\)

We will use the method of factorization to find out the cube root of 2744000 

Factorizing 2744000 into prime factors, 

We get, 

2744000 = 2×2×2×2×2×2×5×5×5×7×7×7 

Now group the factors into triples of equal factors, we get, 

2744000 = (2×2×2) ×(2×2×2) ×(5×5×5) ×(7×7×7) 

As we can see that all the prime factors of 2744000 can be grouped in to triples of equal factors and no factor is left over. 

Now take one factor from each group and by multiplying we get, 

2×2×5×7 = 140 So we can say that 2744000 is a cube of 140

Hence,

\(\sqrt[3]{-2744000}\)

\(=-\sqrt[3]{2744000}\)

\(=-140\)

(iv) We have,

\(\sqrt[3]{-753571}\)

\(=-\sqrt[3]{753571}\)

By using unit digit method, 

Let’s take Number = 753571 

Unit digit = 1 So unit digit in the cube root of 753571 = 

1 After striking out the units, tens and hundreds digits of 753571, 

Now we left with 753.

 As we know that 9 is the Largest number whose cube is less than or equals to 753(9³<753<10³). 

So, The tens digit of the cube root of 753571 is 9.

\(\sqrt[3]{753571} = 91\)

\(\sqrt[3]{-753571}\)

\(=-\sqrt[3]{753571} = -91\)

(v) We have,

\(\sqrt[3]{-32768}\)

\(=-\sqrt[3]{32768}\)

By using unit digit method, we will find out the cube root of 32768, 

Let’s take Number = 32768 

Unit digit = 8 

So unit digit in the cube root of 32768 = 2 

After striking out the units, tens and hundreds digits of 32768, 

Now we left with 32. 

As we know that 9 is the Largest number whose cube is less than or equals to 32(3³<32<4³). So, The tens digit of the cube root of 32768 is 3.

\(\sqrt[3]{32768}=32\)

\(\sqrt[3]{-32768}\)

\(=-\sqrt[3]{32768} = -32\)

A perfect cube can be expressed as a product of three numbers of equal factors

Now by resolving the given number into prime factors we get,

243 = 3 × 3 × 3 × 3 × 3

Since the given number has more than three factors

∴ 243 is not a perfect cube.

The prime factors of 343 = 7 × 7 × 7

Now by grouping into three we get, (7 × 7 × 7)

∴ \(\sqrt[3]{(7)^3}\) = 7

A perfect cube can be expressed as a product of three numbers of equal factors

Now by resolving the given number into prime factors we get,

125 = 5 × 5 × 5

Its cube can be expressed as 5³

∴ 125 is a perfect cube.

(i) -125

-125 = ∛-125 

= -∛125 

= ∛ (5×5×5)

= -5

(ii) -5832

-5832 = ∛-5832 

= -∛5832

Where, unit digit of 5832 = 2

Unit digit in the cube root of 5832 will be 8

After striking out the units, tens and hundreds digits of 5832,

Now we left with 5 only.

As we know that 1 is the Largest number whose cube is less than or equal to 5.

So, the tens digit of the cube root of 5832 is 1.

∛-5832 = -∛5832 

= -18

(iii) -2744000

∛-2744000 = -∛2744000

2744000 = 2×2×2×2×2×2×5×5×5×7×7×7

2744000 = (2×2×2) × (2×2×2) × (5×5×5) × (7×7×7)

= 2×2×5×7 

= 140

∴ ∛-2744000 = -∛2744000 

= -140

\((1\frac{2}{5})^3=(\frac{7}{5})^3=\frac{7\times7\times7}{5\times5\times5}\) = \(\frac{343}{125}.\) 

The prime factors of 64 = 2 × 2 × 2 × 2 × 2 × 2

Now by grouping into three we get, (2 × 2 × 2) × (2 × 2 × 2)

∴ \(\sqrt[3]{((2)^3 × (2)^3)}\) 

= (2 × 2) 

= 4

(i) 8×125  

We know that for any two integers a and b, ∛(a×b) = ∛a × ∛b

By using the property,

∛ (8×125) = ∛8 × ∛125

= ∛(2×2×2) × ∛(5×5×5)

= 2×5

= 10

(ii) -1728×216

We know that for any two integers a and b, ∛(a×b) = ∛a × ∛b

By using the property,

∛(-1728×216) = ∛-1728 × ∛216

We shall use the unit digit method,

Let the number 1728, where Unit digit = 8

The unit digit in the cube root of 1728 will be 2

After striking out the units, tens and hundreds digits of the given number, we are left with the 1.

We know 1 is the largest number whose cube is less than or equal to 1.

So, the tens digit of the cube root of 1728 = 1

∛1728 = 12

Now, let’s find the prime factors for, 

216 = 2×2×2×3×3×3

By grouping the factors in triples of equal factor, we get,

216 = (2×2×2) × (3×3×3)

∛216 = 2×3 

= 6

From above we take as,

∛(-1728×216) = ∛-1728 × ∛216

= -12 × 6

= -72

125 ∛a⁶ – ∛125a⁶

= 125∛(a²)³ – ∛53(a²)³

= 125a² – 5a²

= 120a²

(i) ∛27 × ∛64 

= ∛(27×64)

L.H.S = ∛27 × ∛64

∛27 × ∛64 

= ∛(3×3×3) × ∛(4×4×4)

= 3×4

= 12

R.H.S = ∛(27×64)

∛ (27×64) 

= ∛(3×3×3×4×4×4)

= 3×4

= 12

∴ L.H.S = R.H.S

(ii) ∛ (64×729) 

= ∛64 × ∛729

 L.H.S =  ∛(64×729)

∛(64×729) = ∛(4×4×4×9×9×9)

= 4×9

= 36

R.H.S = ∛64 × ∛729

∛64 × ∛729 = ∛(4×4×4) × ∛(9×9×9)

= 4×9

= 36

∴ L.H.S = R.H.S

(i) ∛ (-125×216) = ∛-125 × ∛216

L.H.S =  ∛(-125×216)

∛ (-125×216) = ∛ (-5×-5×-5×2×2×2×3×3×3)

= -5×2×3

= -30

 R.H.S =  ∛-125 × ∛216

∛-125 × ∛216 = ∛(-5×-5×-5) × ∛(2×2×2×3×3×3)

= -5×2×3

= -30

∴ L.H.S = R.H.S

(ii) ∛(-125×-1000) = ∛-125 × ∛-1000

L.H.S =  ∛ (-125 ×(-1000))

∛ (-125×-1000) = ∛ (-5×(-5)×(-5)×(-10)×(-10)×(-10))

= -5 × -10

= 50

 R.H.S =  ∛-125 × ∛-1000

∛-125 × ∛-1000 = ∛(-5×-5×-5) × ∛(-10×-10×-10)

= -5×-10

= 50

∴ L.H.S = R.H.S

The operation of finding the cube root is the opposite operation of finding the cube.

The prime factors of 216 = 2 × 2 × 2 × 3 × 3 × 3

Now by grouping into three we get, (2 × 2 × 2) × (3 × 3 × 3)

∴ \(\sqrt[3]{ – ((2)^3 × (3)^3)}\)

= – (2 × 3) 

= -6

The unit’s digit of the cube of 18 is 2.

(i) ∛(43 × 63)

We know that for any two integers a and b, ∛(a×b) = ∛a × ∛b

By using the property,

∛ (43 × 63) = ∛43 × ∛63

= 4 × 6

= 24

(ii) ∛(8×17×17×17)

We know that for any two integers a and b, ∛(a×b) = ∛a × ∛b

By using the property,

∛(8×17×17×17) = ∛8 × ∛17×17×17

= ∛23 × ∛173

= 2 × 17

= 34

(iii) ∛(700×2×49×5)

∛(700×2×49×5) = ∛(2×2×5×5×7×2×7×7×5)

= ∛(2³×5³×7³)

= 2×5×7

= 70