Requirements for a Perfect Cube

Oct 13, 2025 by miodeya.com 74 views

Introduction

In mathematics, a perfect cube is a number that can be expressed as the cube of an integer. For example, 27 is a perfect cube because it can be expressed as 3^3. However, not all numbers are perfect cubes. In this article, we will discuss how to find the least number by which a given expression must be multiplied to obtain a perfect cube.

Understanding the Problem

The given expression is $$2^4 \times 3^2 \times 5$$. To find the least number by which this expression must be multiplied to obtain a perfect cube, we need to analyze the prime factorization of the expression.

Prime Factorization

The prime factorization of the expression $$2^4 \times 3^2 \times 5$$ is:

2^4 (2 to the power of 4)
3^2 (3 to the power of 2)
5 (5 to the power of 1)

Requirements for a Perfect Cube

For a number to be a perfect cube, its prime factorization must have each prime factor raised to a power that is a multiple of 3. In other words, the exponent of each prime factor must be a multiple of 3.

Analyzing the Prime Factorization

Let's analyze the prime factorization of the expression $$2^4 \times 3^2 \times 5$$:

2^4: The exponent of 2 is 4, which is not a multiple of 3. To make it a multiple of 3, we need to multiply 2^4 by 2^2 (2 to the power of 2) to get 2^6.
3^2: The exponent of 3 is 2, which is not a multiple of 3. To make it a multiple of 3, we need to multiply 3^2 by 3^1 (3 to the power of 1) to get 3^3.
5: The exponent of 5 is 1, which is not a multiple of 3. To make it a multiple of 3, we need to multiply 5 by 5^2 (5 to the power of 2) to get 5^3.

Calculating the Least Number

Now that we have analyzed the prime factorization of the expression, we can calculate the least number by which it must be multiplied to obtain a perfect cube.

2^4 needs to be multiplied by 2^2 to get 2^6, which is 2^2 (2 to the power of 2) = 4.
3^2 needs to be multiplied by 3^1 to get 3^3, which is 3^1 (3 to the power of 1) = 3.
5 needs to be multiplied by 5^2 to get 5^3, which is 5^2 (5 to the power of 2) = 25.

Conclusion

The least number by which $$2^4 \times 3^2 \times 5$$ must be multiplied to obtain a perfect cube is the product of the numbers calculated above:

4 (2^2) * 3 (3^1) * 25 (5^2) = 300

Therefore, the least number by which $$2^4 \times 3^2 \times 5$$ must be multiplied to obtain a perfect cube is 300.

Example Use Case

Suppose we want to find the least number by which 24 must be multiplied to obtain a perfect cube. The prime factorization of 24 is 2^3 * 3. To make it a perfect cube, we need to multiply 2^3 by 2^3 to get 2^6, and multiply 3 by 3^2 to get 3^3. The least number by which 24 must be multiplied to obtain a perfect cube is 2^3 (8) * 3^2 (9) = 72.

Conclusion

In this article, we discussed how to find the least number by which a given expression must be multiplied to obtain a perfect cube. We analyzed the prime factorization of the expression $$2^4 \times 3^2 \times 5$$ and calculated the least number by which it must be multiplied to obtain a perfect cube. The least number is 300. We also provided an example use case to illustrate the concept.

Introduction

In our previous article, we discussed how to find the least number by which a given expression must be multiplied to obtain a perfect cube. In this article, we will answer some frequently asked questions related to this topic.

Q: What is a perfect cube?

A: A perfect cube is a number that can be expressed as the cube of an integer. For example, 27 is a perfect cube because it can be expressed as 3^3.

Q: How do I determine if a number is a perfect cube?

A: To determine if a number is a perfect cube, you need to check if its prime factorization has each prime factor raised to a power that is a multiple of 3.

Q: What is the prime factorization of an expression?

A: The prime factorization of an expression is the expression broken down into its prime factors. For example, the prime factorization of 24 is 2^3 * 3.

Q: How do I find the least number by which an expression must be multiplied to obtain a perfect cube?

A: To find the least number by which an expression must be multiplied to obtain a perfect cube, you need to analyze the prime factorization of the expression and calculate the least number by which each prime factor must be multiplied to make its exponent a multiple of 3.

Q: What if the exponent of a prime factor is already a multiple of 3?

A: If the exponent of a prime factor is already a multiple of 3, then you do not need to multiply it by anything.

Q: Can you provide an example of finding the least number by which an expression must be multiplied to obtain a perfect cube?

A: Suppose we want to find the least number by which 24 must be multiplied to obtain a perfect cube. The prime factorization of 24 is 2^3 * 3. To make it a perfect cube, we need to multiply 2^3 by 2^3 to get 2^6, and multiply 3 by 3^2 to get 3^3. The least number by which 24 must be multiplied to obtain a perfect cube is 2^3 (8) * 3^2 (9) = 72.

Q: What if the expression has multiple prime factors with different exponents?

A: If the expression has multiple prime factors with different exponents, then you need to analyze each prime factor separately and calculate the least number by which it must be multiplied to make its exponent a multiple of 3.

Q: Can you provide an example of finding the least number by which an expression with multiple prime factors must be multiplied to obtain a perfect cube?

A: Suppose we want to find the least number by which 30 must be multiplied to obtain a perfect cube. The prime factorization of 30 is 2 * 3 * 5. To make it a perfect cube, we need to multiply 2 by 2^2 to get 2^3, multiply 3 by 3^2 to get 3^3, and multiply 5 by 5^2 to get 5^3. The least number by which 30 must be multiplied to obtain a perfect cube is 2^2 (4) * 3^2 (9) * 5^2 (25) = 900.

Q: How do I know if I have found the least number by which an expression must be multiplied to obtain a perfect cube?

A: You can check if you have found the least number by which an expression must be multiplied to obtain a perfect cube by multiplying the expression by the calculated number and checking if the result is a perfect cube.

Conclusion

In this article, we answered some frequently asked questions related to finding the least number by which an expression must be multiplied to obtain a perfect cube. We provided examples and explanations to help you understand the concept.