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Cube Root of 64 is the value that results in the original number 64 after getting multiplied by itself three times. It is the number that gives the product as 64 on multiplication with itself three times.
- The value of cube root of 64 is 4.
- It is expressed as ∛64 in radical form and as (64)⅓ or (64)0.33 in the exponent form.
- Cube root of 64 is the real solution of the equation x3 = 64.
- The number 64 is a perfect cube as the cube root of 64, i.e. 4 is a whole number.
- Since 64 is a two-digit number, the method of prime factorization can be used to calculate the cube root of 64.
- Cube root is the reverse method of evaluating the cube of a number.
Read More: NCERT Solutions for Class 8 Mathematics Cube and Cube Roots
Key Terms: Cube Root of 64, Cube Root, Multiplication, Natural Number, Prime Factorization, Cube, Prime Factors, Irrational Cube Root
What is Cube Root of 64?
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Cube Root of 64 is defined as the number which when multiplied by itself three times gives the product as 64 only. Cube root of 64 is 4. It is represented as 3√64. The number 64 can be expressed as the following in terms of its prime factors:
64 = 2 × 2 × 2 × 2 × 2 × 2
| 64 = 3√(2 × 2 × 2 × 2 × 2 × 2) = 4 |
- Cube Root of 64: 4
- Cube Root of 64 in Exponential Form: (64)1/3 or (64)0.33
- Cube root of 64 in Radical Form: 3√64
Cube Root of 64
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How to Find Cube Root of 64?
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Cube Root of 64 can be calculated by the method of Prime Factorization. If the same factors are present in a pair of three, under the cube root, then they are taken out as a single-digit, since the cube of a number will cancel the cube root.
Here is a step-by-step procedure to calculate the cube root of 64:
Step 1: On the prime factorization of 64, we get
| 64 = 2 × 2 × 2 × 2 × 2 × 2 |
Step 2: Pair up the factors of 64 in a group of three, so that they form cubes:
64 = (2 × 2 × 2) × (2 × 2 × 2)
64 = 23 × 23
Using the exponent rules, we get
64 = 23+3 = 26 [am.an = (a)m+n]
64 = (22)3 [(am)n = amn]
64 = 43
Step 3: Take the cube root on both sides of the above expression like
3√64 = 3√(43)
The cube root is eliminated by the cube of 4. Thus, we get the value of a cubic root of 64, which is
| 3√64 = 4 |
Read More: Cubes and Cube Roots MCQs
Is Cube Root of 64 Irrational?
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Irrational numbers are those numbers that cannot be represented as simple fractions. Irrational Numbers cannot be expressed as a ratio, such as p/q, where p and q are integers, q≠0. To check whether the cube root of a number is rational or irrational, it needs to be checked if the cube root can be represented in p/q form.
| 3√64 = 3√(2 × 2 × 2 × 2 × 2 × 2) = 4 |
Thus, the cube root of 64 is not irrational as it can be expressed in the form of p/q i.e. 4/1. Thus, the value of the cube root of 64 is an integer which means it is rational.
Cube Root of 64
Solved Examples on Cube Root of 64
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Given below are a few examples on the cube root of 64:
Example 1: Calculate the value of ∛64 ÷ ∛(-64).
Solution: It must be noted that the cube root of -64 is equal to the negative of the cube root of 64.
∛-64 = -∛64
Thus,
∛64 ÷ ∛(-64)
= ∛64 ÷ (-∛64)
= 4 ÷ -4 = -1
Thus, the value of ∛64 ÷ ∛(-64) is -1.
Example 2: The volume of a spherical ball is given as 64π in3. What will be the radius of this ball?
Solution: It is given that,
Volume of Spherical Ball = 64π in3
= 4/3 × π × R3
R3 = 3/4 × 64
R = ∛(3/4 × 64) = ∛(3/4) × ∛64
R = 0.90856 × 4 (∵ ∛(3/4) = 0.90856 and ∛64 = 4)
R = 3.63424 in3
Thus, the radius of the given spherical ball is 3.63424 in3.
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Cubes From 1 to 10
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Cube of a number is a value obtained after multiplying a number three times by itself. Cubes of the numbers from 1 to 10 are provided below to find out the cube root easily and assist in 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 |
Things to Remember
- Cube root of 64 is the number that gives the product as 64 on multiplication with itself three times.
- The value of the cube root of 64 is 4.
- Prime factorization is used to calculate the cube root of 64.
- The radical form of the cube root of 64 is ∛64.
- The exponential form of the cube root of 64 is (64)1/3 or (64)0.33.
- 64 is a perfect cube as the cube root of 64 is a whole number (4).
- Cube root of 64 is rational as it can be represented in the p/q form.
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Sample Questions
Ques. What is the value of Cube Root of 64? (2 Marks)
Ans. The number 64 can be expressed in the following manner on prime factorization,
64 = 2 × 2 × 2 × 2 × 2 × 2
∛64 = ∛(2 × 2 × 2 × 2 × 2 × 2) = 4.
Thus, the value of the cube root of 64 is 4.
Ques. Simplify the cube root of 64 divided by the cube root of 125. (3 Marks)
Ans. According to the questions,
Cube Root of 64 / Cube Root of 125 =?
We know that the cube root of 64 is 4 and the cube root of 125 is calculated as 5. Thus,
∛(64/125) = (∛64)/(∛125)
= 4/5 = 0.8
Thus the value of the cube root of 64 divided by the cube root of 125 is 0.8.
Ques. Is 64 a Perfect Cube? (2 Marks)
Ans. On prime factorization, the number 64 gives 2 × 2 × 2 × 2 × 2 × 2.
Combining the prime factors in groups of 3 we get 4 like
64 = ∛(2 × 2 × 2 × 2 × 2 × 2) = 4.
Now, 4 is a whole number which makes 64 a perfect cube.
Ques. If the Cube Root of 64 is 4, what will be the Value of ∛0.064? (2 Marks)
Ans. For the solution, represent ∛0.064 in p/q form i.e.
∛(64/1000) = 4/10 = 0.4.
Thus, the value of ∛0.064 = 0.4.
Ques. Why is Cube Root of 64 Rational? (1 Mark)
Ans. The value of the cube root of 64 is 4 which can be expressed in the form of p/q i.e. = 4/1, where q ≠ 0. Thus, the cube root of 64 is rational.
Ques. Find whether the value of the Cube Root of 64 Minus Cube Root of 8 is a Rational Number. (2 Marks)
Ans. We know that the value of the cube root of 64 is 4 and the value of the cube root of 8 is 2, i.e.
3√64 - 3√8 = 4 - 2 = 2.
Thus, the difference between the cube root of 64 and the cube root of 8 is a rational number (2).
Ques. Find the value of the Cube Root of -64. (3 Marks)
Ans. The cube root of negative numbers is negative.
- First, write the number -64 as a product of its prime factors like -64= -(2 × 2 × 2 × 2 × 2 × 2)
- Pair up the factors into groups of three each like -64= -23 × 23 = -43
- Put the cube root on both sides of the equation: 3√-64 = 3√-43 = -4
Thus, the cube root of -64 is -4.
Ques. What is the value of the Cube Root of 64? (2 Marks)
Ans. According to the question,
(∛64)3 =?
(∛64)3 = 64
Thus, the cube of the cube root of 64 is the number 64 itself.
Ques. Calculate the real root of the equation x3 − 64 = 0. (3 Marks)
Ans. It is given that
x3 − 64 = 0
x3 = 64
Solving for x, we get
x = ∛64
x = ∛64 × (-1 + √3i)/2
x = ∛64 × (-1 - √3i)/2
Where i is called the imaginary unit and is equivalent to √-1.
Ignoring imaginary roots here, we get
x = ∛64
Thus, the real root of the equation x3 − 64 = 0 is for x = ∛64 = 4.
Ques. Is 64 a prime or composite number? (1 Mark)
Ans. 64 is a composite number since the number 64 has more than 2 factors other than 1 and 64 itself. The factors of 64 are 1, 2, 4, 8, 16, 32, and 64.
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