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Cube Root of 729 is 9. The cube root of 729 is a value that after getting multiplied by itself thrice gives the original value. It is expressed as ∛729 in radical form and as (729)⅓ or (729)0.33 in the exponent form. The cube root of 729 is a whole number which means 729 is a perfect cube. For instance, the cube root of 729 is 9. It means when 9 is multiplied by itself thrice it gives the value 720. 9 x 9 x 9 = 729, thus, 9 is the cube root of 729.
Key Terms: Cube Root of 729, Cube Root, Cube, Perfect Cube, Prime Factorization, Whole Number, Cubic Root, Rational Number
Cube Root of 729
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The cube root of a number is a mathematical operation that finds the value that, when raised to the power of 3 (cubed), gives the original number. In other words, if a number x is cubed, the cube root of that result gives the original value of x.
The cube root of a number can be represented using the ∛ symbol, such as ∛27 to represent the cube root of 27. To find the cube root of a number, you can use a calculator, but you can also find it manually by trial and error, or by using a logarithm or exponentiation.
Cube Root of 729 refers to the number or value which when multiplied by itself three times gives the product as 729. The prime factorization of 729 can be expressed as 3 × 3 × 3 × 3 × 3 × 3. Thus, the cube root of 729 is ∛(3 × 3 × 3 × 3 × 3 × 3) = 9.
- The cube root of 729 is 9.
- This is because 9 multiplied by itself three times (9 x 9 x 9) is equal to 729.
- So, the cube root of 729 is the value that, when multiplied by itself three times, gives 729 as the result.
Cube Root of 729
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| Relevant Concepts | ||
|---|---|---|
| Cube and Cube Roots | Cube Root 1 to 30 | Finding Square Roots through Prime Factorization |
| Difference of Cubes Formula | Estimating Square Root | Square Root Formula |
Cube root of 729 by Prime Factorization
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To find the cube root of 729 by the prime factorization method, we need to factorize 729 into its prime factors. The prime factorization method is usually used to find the factors of the given number present under the cubic root. The cubic root cancels the cubed number present within it.
First, we need to find the prime factors of 729, and then pair them in a group of three to give us the cube of factors. After finding the cube of factors of the given number, we apply the cube root, which gets cancelled with the cubes.
The steps to calculate the cube root of 729 are
Step 1: First, find the prime factors of 729.
729 = 3 × 3 × 3 × 3 × 3 × 3
Step 2: It is clearly seen that 729 is a perfect cube. Using the laws of exponents – a(m x n) = (am)n, we can simplify this to:
729 = (3³ x 3³)(1/3)
Using (a x b)n = an x bn, we can simplify this to:
729 = 3(3 x 1/3) x 3(3 x 1/3)
729 = [32]
3√729 = 9
Thus, 3√729 = 9.
Cube root of 729 by prime factorization
Is Cube Root of 729 Irrational?
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No, the cube root of 729 is not irrational.
∛729 = ∛(3 × 3 × 3 × 3 × 3 × 3)
It can be expressed in the form of p/q which is 9/1. Thus, the value of the cube root of 729 is an integer which means it is rational.
Solved Examples on Cube Root of 729Example 1: Calculate the value of ∛729 ÷ ∛(-729). Solution: The cube root of -729 is equivalent to the negative of the cube root of 729. ∛-729 = -∛729 ∛729 / ∛(-729) = ∛729 / (-∛729) = -1 Example 2: The volume of a cube is 729 in3. What will be the length of the side of the cube? Solution: It is given that the volume of the cube is Volume of the Cube = 729 in3 = a3 a3 = 729 If we Cube root on both sides of the equation, we get a = ∛729 inch The cube root of 729 is 9, thus, the length of the side of the cube is 9 inch. |
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|---|---|---|
| Square Root Formula | Square Roots | Root Mean Square |
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Cube Root of Perfect Cubes
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A perfect cube is defined as a number that is obtained by multiplying the same integer three times. The cube root of perfect cubes up to three-digit numbers is easy and should be memorized for faster calculations.
| Number (n) | Cubes (n3) |
|---|---|
| 1 | 1 |
| 2 | 8 |
| 3 | 27 |
| 4 | 64 |
| 5 | 125 |
| 6 | 216 |
| 7 | 343 |
| 8 | 512 |
| 9 | 729 |
| 10 | 1000 |
The cube root is a mathematical operation that determines the value that, when multiplied by itself three times, gives the original number. In this case, 9 x 9 x 9 equals 729. The cube root of 729 is commonly used in math and science to calculate various formulas and equations. It's a fundamental concept that helps us understand the relationship between numbers and their roots.
Things to Remember
- Cube Root of 729 is 9.
- Cube Root of 729 is the value that results in the number 729 on multiplication with itself thrice.
- It can be expressed as 9 x 9 x 9 = 729.
- The radical form of the cube root of 729 is ∛729.
- The exponential form of the cube root of 729 is (729)⅓ or (729)0.33.
- 729 is a perfect cube which means that the cube root of 729 is a whole number (9).
Sample Questions
Ques. What will be the Cube Root of Product of Integers ∛(729 × 125)? (3 Marks)
Ans. We have (∛729 × ∛125).
Removing the brackets, we get
= ∛729 × ∛125
∛9×9×9 × ∛5×5×5
= (9 × 5)
= 45
Thus, the Cube Root of Product of Integers (∛729 × ∛125) is 45.
Ques. Calculate the real root of the equation x3 − 729 = 0. (3 Marks)
Ans. x3 − 729 = 0
It means that x3 = 729
Solving for x,
x = ∛729
x = ∛729 × (-1 + √3i))/2 and x = ∛729 × (-1 - √3i))/2, where i is the imaginary unit and is equal to √-1.
Ignoring imaginary roots,
x = ∛729
Thus, the real root of the equation x3 − 729 = 0 is for x = ∛729 = 9.
Ques. Find out the value of the Cube Root of 729. (2 Marks)
Ans. The cube root of 729 can be expressed as
729 = 3 × 3 × 3 × 3 × 3 × 3
∛729 = ∛(3 × 3 × 3 × 3 × 3 × 3) = 9.
Thus, the value of the cube root of 729 is 9.
Ques. Find the value of ∛0.729 if the Cube Root of 729 is 9. (2 Marks)
Ans. First we will represent ∛0.729 in p/q form
∛(729/1000) = 9/10
= 0.9.
Thus, the value of ∛0.729 is 0.9.
Ques. Is 729 a Perfect Cube? (2 Marks)
Ans. On prime factorization, the number 729 gives
3 × 3 × 3 × 3 × 3 × 3
Combining the prime factors in groups of 3 gives 9. Thus, the cube root of 729 = ∛(3 × 3 × 3 × 3 × 3 × 3) = 9. Hence 729 is a perfect cube.
Ques. Is the value of the Cube Root of 729 Rational? (2 Marks)
Ans. Yes, the value of the cube root of 729 is rational. It is so because the value of the cube root of 729 can be expressed in the form of p/q i.e. = 9/1, where q ≠ 0. Hence, the ∛729 is rational.
Ques. What is the Cube of the Cube Root of 729? (2 Marks)
Ans. We know that the cube root of 729 is 9. Now, the cube of 9 is 729 itself. Thus, the cube of the cube root of 729 is the number 729 itself.
(∛729)3 = (7291/3)3 = 729
Ques. How to find the cube root of a number? (2 Marks)
Ans. The cube root of a number can be found by a very simple and easy method which is the prime factorization method. The cube root of a number is denoted by '∛ ' symbol.
For instance, ∛8 = ∛(2 × 2 × 2) = 2.
It shows that 8 is a perfect cube number.
Ques. How to find the cube root of 729 by prime factorization? (3 Marks)
Ans. To find the cube root of 729 through prime factorization, follow the given steps:
- Prime factorization of 729 is 3 × 3 × 3 × 3 × 3 × 3.
- On simplifying the above expression, we get 36
- On further simplification, 93
- Thus, the cube root of 729 by prime factorization is (3 × 3 × 3 × 3 × 3 × 3)1/3 = 9.
Ques. Is 729 a perfect square or not? (2 Marks)
Ans. Yes, 729 is a perfect square.
27 x 27 = 729
Thus, 729 is a perfect square number, and 729 is a square of 27.
Ques. Is the square root of 729 irrational or rational? (2 Marks)
Ans. 729 is a perfect square which means its square root will have no decimal. The square root of 729 is 27. It is a whole number and can be represented in the form of p/q, thus, it is a rational number.
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