Cube Root of 1728: Prime Factorisation and Estimation Method

Collegedunia Team

Content Curator

Cube root of a number is the resultant number when a number is multiplied three times by itself. It is denoted as 3x.

1728 can be expressed as the product of many integers, like as 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3.
The resulting cube root of 1728 = ∛(2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3) = 12.
So, there are two methods used to find out the cube root of any particular number
Prime factorization method.
Estimation method.

Table of Content

Cube Root of 1728
Perfect cube
Prime Factorisation Method to Find Out Cube Root of 1728
Estimation Method to Find Out the Cube Root of 1728
Things to Remember
Sample Questions

Key Terms: Cube root, Square, Cube, Factorization, Estimation.

Cube Root of 1728

[Click Here for Sample Questions]

The cube root of 1728 is denoted, 31728, and when it is solved, the resulting number is multiplied three times to get 1728.

There are two methods to find out the value of 31728 which are - the estimation method and the prime factorization method.
1728 can be written in the form of multiplication of numbers, such as – 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3.
The resulting cube root of 1728 = ∛(2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3) = 12.

Read More: Cube Root of 729

Perfect Cube

[Click Here for Sample Questions]

To find out if a number is a perfect cube, prime factors are found and then grouped into triplets. If none of the digits is left alone, then the number is a perfect cube. With an example, this concept can be understood completely-

Assume there is a number "125.” To find out if is a perfect cube, first of all, prime factors are found 5*5*5 = 125
There is one triplet of "5" in this equation, and none of the digits are left alone, so 125 is a perfect cube.

Read More: Cubes from 1 to 50

Prime Factorisation Method to Find Out Cube Root of 1728

[Click Here for Sample Questions]

1728 is a perfect cube. It is a four-digit number, so solving it with the prime factorization method is a little lengthy. Click here to learn how to find prime factors.

Step 1: First of all, we have to find out the prime factor of 1728

When solved, the prime factor of 1728 will be 2*2*2*2*2*2*3*3*3

Step 2: Now all these factors will be grouped in a pair of three digits

1728 = (2*2*2)(2*2*2)(3*3*3)

Step 3: The above sum can be grouped as -

1728 = 2³*2³*3³

Step 4: Now, applying cube root on both sides of the equation -

∛1728 = ∛2³ *2³ *3³

Step 5: Solving the above equation -

2*2*3 = 12

That's why

∛1728 = 12

Cube root is neutralized by cube root, so the cube root of 1728 will be 12.

Read More: Cube and Cube Roots

Estimation Method to Find Out the Cube Root of 1728

[Click Here for Sample Questions]

To solve a cube root by the estimation method, one must memorize the cube root table.

Step 1: First and foremost, check the number's unit place. Here, for 1728, the unit place digit is 8.

Step 2: Now, from the cube root table, you will have to check which number has 8 as its unit digit. The cube root of 2 is 8 or 2³ = 8

This indicates that the unit place of the cube root of 1728 will be “2.”

Step 3: Now, for a moment, forget about the last three digits of 1728, the remaining digit will be “1.”

Step 4: Find out the cube root of the remaining number. In this case, the remaining number is 1 and the cube root of 1 is 1.

One obtained two digits, 1 and 2, by following these steps. According to step 2, the digit “2” will be on the unit digit. So, the answer will be 12.

Therefore, ∛1728 = 12.

Read More: Cubes and Cube Roots MCQs

Things to Remember

The cube root of 1728 is denoted, ∛1728.
1728 can be written in the form of multiplication of numbers, such as – 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3.
The resulting cube root of 1728 = ∛(2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3) = 12.
1728 is a perfect cube.
For 1728, the unit place digit is 8.
There are two methods used to find out the cube root of any particular number – the Prime factorization method and the Estimation method.
To find out if a number is a perfect cube, prime factors are found and then grouped into triplets.

Sample Questions

Ques: Use the prime factorization method to find out the cube root of 13824. (4 Marks)

Ans: To find out the cube root of a particular number, first of all, you will have to find out the prime factor of the number.

Here, find the prime factor of 13824. It will be -

13824 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3

Now all these factors will be grouped in a pair of three digits

13824 = (2*2*2)(2*2*2)(2*2*2) (3*3*3)

Step 3: The above sum can be grouped as -

13824 = 2³*2³*2³*3³

Step 4: Now, applying cube root on both sides of the equation -

∛13824 = ∛2³*2³*2³*3³

Step 5: Solving the above equation -

2*2*2*3 = 24

Both sides' cube roots will be neutralized.

Therefore, ∛13824 = 24

Ques: Use the prime factorization method to find out the cube root of 3375. (4 Marks)

Ans: Here, find the prime factor of 13824. It will be -

3375 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3

Now all these factors will be grouped in a pair of three digits

3375 = (3*3*3)(5*5*5)

Step 3: The above sum can be grouped as -

3375 = 3³*5³

Step 4: Now, applying cube root on both sides of the equation -

∛3375 = ∛3³*5³

Step 5: Solving the above equation -

3*5 = 15

Both sides' cube roots will be neutralized.

Therefore,∛3375 = 15

Ques: Use the prime factorization method to find out the cube root of 46656. (4 Marks)

Ans: First of all, make a group of three-digit numbers from the right side.

In this case, 46656, the three-digit group from the right side is "656,” and the remaining number, here "46,” will fall in the second group.

46656 is ending with "6,” so the unit digit will be 6.

Therefore, the unit digit of the cube root of 46656 is 6. - (i)

Now, check the second group; here, “46” is falling between the cube roots of which two numbers.

46 is greater than 27 (33= 27)

46 is smaller than 64 (43= 64)

33 < 46 < 43

Smaller number in the above equation is 3

So, the tens digit of the cube root of 46656 is 3 - (ii)

Now, from (i) and (ii) equation, the one's digit is 6 and the tens digit is 3

Therefore, ∛46656 = 36

Ques: Use the prime factorization method to find out the cube root of 91125. (4 Marks)

Ans: First of all, make a group of three-digit numbers from the right side.

For 91125, the three-digit group from the right side is "125,” and the remaining number, here "91,” will fall in the second group.

91125 is ending with "5", so the unit digit will be 5.

Therefore, the unit digit of the cube root of 91125 is 5. - (i)

Now, check the second group; here, “46” is falling between the cube roots of which two numbers.

91 is greater than 64 (43= 27)

91 is smaller than 125 (53= 125)

43 < 91 < 53

The smaller number in the above equation is 4

So, the tens digit of the cube root of 91125 is 4 - (ii)

Now, from the (i) and (ii) equations, the one's digit is 5 and the tens digit is 4

Therefore, ∛91125 = 45.

Ques: What is the value of ∛1728 ÷ ∛(-1728)? (2 Marks)

Ans: The cube root of -1728 is equal to the negative of the cube root of 1728.

⇒ ∛-1728 = -∛1728

Therefore,

⇒ ∛1728/∛(-1728) = ∛1728/(-∛1728) = -1

Ques: Is 1728 a Perfect Cube? (2 Marks)

Ans: The prime factorization of 1728 gives 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3. Combining the prime factors in groups of 3 gives 12. So, the cube root of 1728 = ∛(2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3) = 12 (perfect cube).

Ques: Why is the value of the Cube Root of 1728 Rational? (2 Marks)

Ans: The value of 1728's cube root may be written as follows: p/q i.e. = 12/1, where q ≠ 0. Therefore, ∛1728 is rational.

Ques: If the Cube Root of 1728 is 12, Find the Value of ∛1.728. (1 Mark)

Ans: By presenting ∛1.728 in p/q form i.e. ∛(1728/1000) = 12/10 = 1.2. Hence, the value of ∛1.728 = 1.2.

For Latest Updates on Upcoming Board Exams, Click Here: https://t.me/class_10_12_board_updates

Check-Out:

Study Guides
NCERT Solutions Class 6 to 12	NCERT Solutions for Class 8 Maths	NCERT Solutions for Class 8 Science
Biology Study Notes	Biology MCQs	Topics for Comparison in Physics
Physics Study Notes	Choice-based questions in physics	Topics with relation in physics
Physics MCQs	Formulas in Physics	Important Physics Constants and Units
Chemistry Study Notes	Comparison topics in Chemistry	Chemistry MCQs
Maths MCQs	Important Chemistry Formulas	Important Named Reactions
Comparison Topics in Maths	Important Maths Formulas	Comparison Topics in Biology

Cube Root of 1728: Prime Factorisation and Estimation Method

Cube Root of 1728

Perfect Cube

Prime Factorisation Method to Find Out Cube Root of 1728

Estimation Method to Find Out the Cube Root of 1728

Things to Remember

Sample Questions

CBSE X Related Questions

1.
Prove that :
\(\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta\).

2.
The graph of \(y = f(x)\) is given. The number of zeroes of \(f(x)\) is :

4.
PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If \(OP = 13\) cm, then find the length AB and PA.

5.
The natural number 1 is :

6.
If the pair of linear equations : \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \) is consistent and dependent, then

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

Cube Root of 1728: Prime Factorisation and Estimation Method

Cube Root of 1728

Perfect Cube

Prime Factorisation Method to Find Out Cube Root of 1728

Estimation Method to Find Out the Cube Root of 1728

Things to Remember

Sample Questions

CBSE X Related Questions

1.Prove that : \(\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta\).

2.The graph of \(y = f(x)\) is given. The number of zeroes of \(f(x)\) is :

4.PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If \(OP = 13\) cm, then find the length AB and PA.

5.The natural number 1 is :

6.If the pair of linear equations : \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \) is consistent and dependent, then

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Prove that :
\(\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta\).

2.
The graph of \(y = f(x)\) is given. The number of zeroes of \(f(x)\) is :

4.
PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If \(OP = 13\) cm, then find the length AB and PA.

5.
The natural number 1 is :

6.
If the pair of linear equations : \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \) is consistent and dependent, then