1. Calculate The Least Number Of Pieces Obtained By Finding The Prime Factors Of 1728 And 2025. Hence, Evaluate: $\[ \frac{\sqrt{1728}}{\sqrt{2025}} \\]2. A Retailer Bought A Bag Of Tea Leaves. If The Retailer Were To Repack The Tea

Introduction

In this article, we will delve into two distinct mathematical problems. The first problem involves finding the prime factors of two given numbers, 1728 and 2025, and then calculating the least number of pieces obtained by dividing these numbers. The second problem requires us to evaluate an expression involving square roots. We will break down each problem step by step and provide a detailed solution.

Problem 1: Finding Prime Factors and Calculating the Least Number of Pieces

Step 1: Find the Prime Factors of 1728 and 2025

To find the prime factors of a number, we need to express it as a product of prime numbers. Let's start by finding the prime factors of 1728 and 2025.

Prime Factors of 1728

The prime factorization of 1728 is:

1728 = 2^6 × 3^3

Prime Factors of 2025

The prime factorization of 2025 is:

2025 = 3^4 × 5^2

Step 2: Calculate the Least Number of Pieces

Now that we have the prime factors of both numbers, we can calculate the least number of pieces obtained by dividing 1728 by 2025.

To do this, we need to find the greatest common divisor (GCD) of the two numbers. The GCD is the largest number that divides both numbers without leaving a remainder.

Calculating the GCD

The GCD of 1728 and 2025 is:

GCD(1728, 2025) = 3^3 = 27

Now that we have the GCD, we can calculate the least number of pieces obtained by dividing 1728 by 2025:

Least Number of Pieces = 1728 ÷ 2025 = 1728 ÷ (3^3 × 5^2) = 1728 ÷ 27 = 64

Step 3: Evaluate the Expression

Now that we have the least number of pieces, we can evaluate the expression:

$\frac{\sqrt{1728}}{\sqrt{2025}}$

To evaluate this expression, we need to find the square roots of both numbers and then divide them.

Evaluating the Square Roots

The square root of 1728 is:

√1728 = 2^3 × 3^2 = 12 × 9 = 108

The square root of 2025 is:

√2025 = 3^4 × 5 = 81 × 5 = 405

Now that we have the square roots, we can evaluate the expression:

$\frac{\sqrt{1728}}{\sqrt{2025}} = \frac{108}{405} = \frac{4}{15}$

Conclusion

In this article, we solved two distinct mathematical problems. The first problem involved finding the prime factors of two given numbers, 1728 and 2025, and then calculating the least number of pieces obtained by dividing these numbers. The second problem required us to evaluate an expression involving square roots. We broke down each problem step by step and provided a detailed solution.

Problem 2: Evaluating an Expression Involving Square Roots

Step 1: Evaluate the Square Roots

The problem requires us to evaluate the expression:

$\frac{\sqrt{1728}}{\sqrt{2025}}$

We already evaluated this expression in the previous problem. The square root of 1728 is:

√1728 = 2^3 × 3^2 = 12 × 9 = 108

The square root of 2025 is:

√2025 = 3^4 × 5 = 81 × 5 = 405

Step 2: Evaluate the Expression

Now that we have the square roots, we can evaluate the expression:

$\frac{\sqrt{1728}}{\sqrt{2025}} = \frac{108}{405} = \frac{4}{15}$

Conclusion

In this article, we solved two distinct mathematical problems. The first problem involved finding the prime factors of two given numbers, 1728 and 2025, and then calculating the least number of pieces obtained by dividing these numbers. The second problem required us to evaluate an expression involving square roots. We broke down each problem step by step and provided a detailed solution.

Discussion

The problems presented in this article are designed to test the reader's understanding of prime factorization, greatest common divisors, and square roots. The first problem requires the reader to find the prime factors of two given numbers and then calculate the least number of pieces obtained by dividing these numbers. The second problem requires the reader to evaluate an expression involving square roots.

Conclusion

In conclusion, this article has provided a detailed solution to two distinct mathematical problems. The first problem involved finding the prime factors of two given numbers, 1728 and 2025, and then calculating the least number of pieces obtained by dividing these numbers. The second problem required us to evaluate an expression involving square roots. We broke down each problem step by step and provided a detailed solution.

References

• [1] "Prime Factorization" by Math Open Reference
• [2] "Greatest Common Divisor" by Math Open Reference
• [3] "Square Roots" by Math Open Reference

Appendix

The following is a list of formulas and theorems used in this article:

• Prime Factorization: A prime factorization of a number is a product of prime numbers that equals the number.
• Greatest Common Divisor: The greatest common divisor of two numbers is the largest number that divides both numbers without leaving a remainder.
• Square Roots: The square root of a number is a value that, when multiplied by itself, equals the number.

Introduction

In our previous article, we solved two distinct mathematical problems. The first problem involved finding the prime factors of two given numbers, 1728 and 2025, and then calculating the least number of pieces obtained by dividing these numbers. The second problem required us to evaluate an expression involving square roots. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on the topics covered.

Q&A

Q: What is prime factorization?

A: Prime factorization is the process of expressing a number as a product of prime numbers. For example, the prime factorization of 12 is 2^2 × 3.

Q: How do I find the prime factors of a number?

A: To find the prime factors of a number, you can use the following steps:

1. Divide the number by the smallest prime number, which is 2.
2. If the number is divisible by 2, continue dividing by 2 until it is no longer divisible.
3. Take the result and divide it by the next prime number, which is 3.
4. Continue this process until you have expressed the number as a product of prime numbers.

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor of two numbers is the largest number that divides both numbers without leaving a remainder. For example, the GCD of 12 and 18 is 6.

Q: How do I calculate the GCD of two numbers?

A: To calculate the GCD of two numbers, you can use the following steps:

1. List the factors of each number.
2. Identify the common factors.
3. Take the largest common factor, which is the GCD.

Q: What is a square root?

A: A square root of a number is a value that, when multiplied by itself, equals the number. For example, the square root of 16 is 4, because 4 × 4 = 16.

Q: How do I evaluate an expression involving square roots?

A: To evaluate an expression involving square roots, you can follow these steps:

1. Simplify the expression by combining like terms.
2. Take the square root of each term.
3. Multiply the square roots together.

Q: Can you provide an example of how to evaluate an expression involving square roots?

A: Let's consider the expression:

$\frac{\sqrt{1728}}{\sqrt{2025}}$

To evaluate this expression, we can follow the steps outlined above:

1. Simplify the expression by combining like terms: $\frac{\sqrt{1728}}{\sqrt{2025}} = \frac{108}{405}$
2. Take the square root of each term: $\sqrt{108} = 6\sqrt{3}$ and $\sqrt{405} = 9\sqrt{5}$
3. Multiply the square roots together: $\frac{6\sqrt{3}}{9\sqrt{5}} = \frac{2\sqrt{3}}{3\sqrt{5}}$

Q: What is the least number of pieces obtained by dividing 1728 by 2025?

A: To find the least number of pieces obtained by dividing 1728 by 2025, we need to find the greatest common divisor (GCD) of the two numbers. The GCD of 1728 and 2025 is 27. Therefore, the least number of pieces obtained by dividing 1728 by 2025 is 1728 ÷ 2025 = 1728 ÷ (3^3 × 5^2) = 1728 ÷ 27 = 64.

Q: Can you provide additional examples of how to find the prime factors of a number?

A: Let's consider the number 24. To find the prime factors of 24, we can follow these steps:

1. Divide 24 by the smallest prime number, which is 2.
2. Continue dividing by 2 until it is no longer divisible: 24 ÷ 2 = 12, 12 ÷ 2 = 6, 6 ÷ 2 = 3.
3. Take the result and divide it by the next prime number, which is 3: 3 ÷ 3 = 1.
4. Express the number as a product of prime numbers: 24 = 2^3 × 3.

Q: Can you provide additional examples of how to calculate the GCD of two numbers?

A: Let's consider the numbers 12 and 18. To calculate the GCD of 12 and 18, we can follow these steps:

1. List the factors of each number: 12 = 1, 2, 3, 4, 6, 12 and 18 = 1, 2, 3, 6, 9, 18.
2. Identify the common factors: 1, 2, 3, 6.
3. Take the largest common factor, which is the GCD: GCD(12, 18) = 6.

Conclusion

In this article, we provided a Q&A section to help clarify any doubts and provide additional information on the topics covered. We hope that this article has been helpful in understanding the concepts of prime factorization, greatest common divisors, and square roots. If you have any further questions or need additional clarification, please don't hesitate to ask.

References

• [1] "Prime Factorization" by Math Open Reference
• [2] "Greatest Common Divisor" by Math Open Reference
• [3] "Square Roots" by Math Open Reference

Appendix

The following is a list of formulas and theorems used in this article:

• Prime Factorization: A prime factorization of a number is a product of prime numbers that equals the number.
• Greatest Common Divisor: The greatest common divisor of two numbers is the largest number that divides both numbers without leaving a remainder.
• Square Roots: The square root of a number is a value that, when multiplied by itself, equals the number.

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