Question

$solve \: it \: properly \: with \: rough \: work \: \\ please \: it \: is \: urgent$
​

Answer 1

1.) Answer → The possible digit at the ones place in the cube root of 9,70,299 is – (c) 9

Therefore, 9 is the possible digit at ones place in the cube root of 9,70,299 –

⟶ As you can see that the provided (number / digit) given in the question = 9,70,299

and Here we are asked to find its possible digit at ones place, so letter start coming from the right side :

9, 70, 299 [ counting from right side]

⟹ 9 × 9 × 9 = (9³) = 729

Therefore, counting from the right side first place we will get the answer as 9.

Hence, shown with justification ✓

━━━━━━━━━━━━━━━━━━━━━━━━━

2.) Answer → The smallest natural number by which 3,072 has to be divided to make it as a perfect cube is – (a) 3

Solution : Now, 3 is the smallest natural number by which 3,072 has to be divided to make it as perfect cube –

⟶ First of all we will have to divide 1024 from 3 [ as per doing pairing of 10 and 24 (each in 2 pairs/ taking the pairs as two]

After that we'll get 62 when (3 and 3) is added at one side and 2 will be written as here it was written in 32 [ taking 2 as common in 62 ]. Therefore, 3 will be the smallest natural number by which 3,072 has to be divided to make it as perfect cube.

━━━━━━━━━━━━━━━━━━━━━━━━━

3.) Answer → Now, as for the provided question we have find the cube root of 8, 30,584 is –

(b) 94

$⟹ \: \sqrt{830584}$

$⟹ \sqrt[3]{830584} = 94 \times 94 \times 94$

$⟹ \: {94}^{3} = (94 \times 94 \times 94)$

$⟹ \: \sqrt{830584} = 94 \times 94 \times 94 \: = ( {94}^{3})$

Therefore, The cube root of 8,30,584 = 94

━━━━━━━━━━━━━━━━━━━━━━━━━

4.) Answer → Here, as per the provided question we are asked here to find the perfect cube from which when 40,000 is divided to make it as perfect cube – (b) 40

Here, 40 will be the perfect cube as it will divide 40,000 very nicely without any problem...

⟶ when we will start dividing by each number they will not be divisible but when we start dividing it with 40 it can be divisible not at the spot but after 40×2 the process used to be continued and therefore 40,000 is divided to make it as perfect cube with help of a number '40'.

━━━━━━━━━━━━━━━━━━━━━━━━━

5.) Answer → Now, here we are said to find the value of ³√-25 × 40 is – (c) -10

$⟹ \: \sqrt[3]{ - 25 \times 40}$

$⟹ \: \sqrt[3]{ - 1000}$

$⟹ \: - 10 \: \times - 10 \times - 10 = {10}^{3}$

$⟹ \: \sqrt[3]{ - 25 \times 40} = - 10$

Therefore, the value of ³√-25 × 40 = -10

━━━━━━━━━━━━━━━━━━━━━━━━━

Answer 2

Required Answers:

$\rm\red{ Q.1}$ The possible digit the ones place in the cube root of 9,70,299 is:

• a) 3
• b) 7
• c) 9 $\large ☑$
• d) 6

$\rm\red{ Q.2}$ The smallest number by which 3072 has to be divided to make it a perfect cube is

• a) 3
• b) 4
• c) 2
• d) 6 $\large ☑$

$\rm\red{ Q.3}$ The cube root of 8,30,584 is:

• a) 92
• b) 94$\large ☑$
• c) 96
• d) 98

$\rm\red{ Q.4}$ The number by which 40,000 should be divided to make it a perfect cube is:

• a) 100
• b) 40 $\large ☑$
• c) 4
• d) 100

$\rm\red{ Q.5}$The value of $\rm{ \sqrt[3]{ - 25 \times 40} }$ is:

• a) 20
• b) 10
• c) -10 $\large ☑$
• d) 5

Answers with Explanation:

$\large\rm { ☆_!! Question !_! ☆}$

$\rm\red{ Q.1}$ The possible digit the ones place in the cube root of 9,70,299 is:

$\large\rm { ☆_!! Solution !_! ☆}$

$\implies$ Whenever this type of questions are asked then we have to only notice the unit/ones digit of the given number.

$\large \rm\red { Given \: number}$ $\large\rm{= 9,70,29 \underline{9} }$

Here the unit digit of the given number is 9, so the possible digit at the ones place in the cube root of 9,70,299 is 9.

And this is because “unit digit of a cube number is the unit digit of its last digit.”

Verification: ( Not needed )

We can also verify it by finding its cube root.To find its cube root, we will resolve it into prime factors-

$\qquad$ 9 | 9,70,299

$\qquad$ 9 | 1,07,811

$\qquad$ 9 | 11,979

$\qquad$ 11 | 1,331

$\qquad$ 11 | 121

$\qquad$ 11 | 11

$\qquad$ | 1

☞ 9,70,299 = 9 × 9 × 9 × 11 × 11 × 11

☞ $\rm{ \sqrt[3]{ 9,70,299} }$ = 9 × 11 = 99

Here, we can see that unit digit of cube root of 9,70,299 is 9. Then, hence verified!!!

${\large {\boxed {\rm {\purple { Option \: C ‼} }}}}$

_________________

$\large\rm { ☆_!! Question !_! ☆}$

$\rm\red{ Q.2}$ The smallest number by which 3072 has to be divided to make it a perfect cube is:

$\large\rm { ☆_!! Solution !_! ☆}$

To find the smallest number by which 3072 has to be divided to make it a perfect cube , we will resolve it into prime factors by by prime factorization-

Resolving 3072 into prime factors-

$\qquad$ 2 | 3072

$\qquad$ 2 | 1536

$\qquad$ 2 | 768

$\qquad$ 2 | 384

$\qquad$ 2 | 192

$\qquad$ 2 | 96

$\qquad$ 2 | 48

$\qquad$ 2 | 24

$\qquad$ 2 | 12

$\qquad$ 2 | 6

$\qquad$ 3 | 3

$\qquad$ | 1

☞ 3072 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3

Clearly , to make 3072 a perfect cube, it must be divided by (2 × 3) = 6 Ans.

${\large {\boxed {\rm {\purple { Option \: D ‼} }}}}$

_____________

$\large\rm { ☆_!! Question !_! ☆}$

$\rm\red{ Q.3}$ The cube root of 8,30,584 is:

$\large\rm { ☆_!! Solution !_! ☆}$

To find the cube root of 8,30,584 , we will resolve it into prime factors,

Resolving 8,30,584 into prime factors, we get -

$\qquad$ 2 | 8,30,584

$\qquad$ 2 | 4,15,292

$\qquad$ 2 | 207646

$\qquad$47| 103823

$\qquad$47| 2209

$\qquad$47| 47

$\qquad$47| 1

☞ 8,30,584 = 2 × 2 × 2 × 47 × 47 × 47

☞$\rm{ \therefore \sqrt[3]{ 8,30,584} }$ = 2 × 47 = 94 ( Answer) } [/tex]

${\large {\boxed {\rm {\purple { Option \: B ‼} }}}}$

_________________

$\large\rm { ☆_!! Question !_! ☆}$

$\rm\red{ Q.4}$ The number by which 40,000 should be divided to make it a perfect cube is:

$\large\rm { ☆_!! Solution !_! ☆}$

To find the number by which 40,000 should be divided to make it a perfect cube , we will resolve it into prime factors-

Resolving 40,000 into prime factors, we get -

$\qquad$ 2 | 40,000

$\qquad$ 2 | 20,000

$\qquad$ 10| 10,000

$\qquad$ 10| 1,000

$\qquad$ 10| 100

$\qquad$ 10| 10

$\qquad$ | 1

☞ 40,000 = 2 × 2 × 10 × 10 × 10 × 10 × 10

Clearly , to make 40,000 a perfect cube, it must be divided by ( 2 × 2 × 10 ) = 40 ( Ans)

${\large {\boxed {\rm {\purple { Option \: B ‼} }}}}$

________________

$\large\rm { ☆_!! Question !_! ☆}$

$\rm\red{ Q.5}$The value of $\rm{ \sqrt[3]{ - 25 \times 40} }$ is:

$\large\rm { ☆_!! Solution !_! ☆}$

We can also write $\rm{ \sqrt[3]{ - 25 \times 40} }$ as $\rm{ \sqrt[3]{ -1000} }$ because -25 × 40 is -1000.

Now to find the cube root of -1000, we will resolve it into prime factors-

Resolving -1000 into prime factors we get,

$\qquad$ 10 | 1000

$\qquad$ 10 | 100

$\qquad$ 10 | 10

$\qquad$ | 1

$\rm { ☞ -1000 = (-10) \times (-10) \times (-10)}$

$\rm{ \therefore \sqrt[3]{ -1000} }$ = -10( Answer)

${\large {\boxed {\rm {\purple { Option \: C ‼} }}}}$