Simplify The Numerical Square Root Expression: $\sqrt{\frac{576}{64}}$1. The Prime Factorization Of 576 Is $\square$2. The Prime Factorization Of 64 Is $\square$3. The Expression 576 64 \sqrt{\frac{576}{64}} 64 576 ​ ​ In Simplest

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Introduction

In mathematics, simplifying numerical expressions is an essential skill that helps in solving complex problems efficiently. One of the common numerical expressions that require simplification is the square root of a fraction. In this article, we will focus on simplifying the numerical square root expression: $\sqrt{\frac{576}{64}}$. We will break down the problem into smaller steps, starting with the prime factorization of the numerator and denominator.

Prime Factorization of 576

To simplify the expression $\sqrt{\frac{576}{64}}$, we need to find the prime factorization of both the numerator (576) and the denominator (64). The prime factorization of a number is the expression of that number as the product of its prime factors.

The Prime Factorization of 576 is 2^6 * 3^2

The prime factorization of 576 can be found by dividing the number by its prime factors. We start by dividing 576 by 2, which gives us 288. Dividing 288 by 2 gives us 144, and dividing 144 by 2 gives us 72. Dividing 72 by 2 gives us 36, and dividing 36 by 2 gives us 18. Dividing 18 by 2 gives us 9, and dividing 9 by 3 gives us 3. Therefore, the prime factorization of 576 is 2^6 * 3^2.

Prime Factorization of 64

Similarly, we need to find the prime factorization of the denominator (64).

The Prime Factorization of 64 is 2^6

The prime factorization of 64 can be found by dividing the number by its prime factors. We start by dividing 64 by 2, which gives us 32. Dividing 32 by 2 gives us 16, and dividing 16 by 2 gives us 8. Dividing 8 by 2 gives us 4, and dividing 4 by 2 gives us 2. Therefore, the prime factorization of 64 is 2^6.

Simplifying the Expression

Now that we have the prime factorization of both the numerator and the denominator, we can simplify the expression $\sqrt{\frac{576}{64}}$.

Simplifying the Expression using Prime Factorization

We can simplify the expression by canceling out the common factors in the numerator and the denominator. In this case, both the numerator and the denominator have a factor of 2^6. We can cancel out this common factor to get:

$\sqrt{\frac{2^6 * 3^2}{2^6}}$

Canceling out the Common Factor

We can cancel out the common factor of 2^6 in the numerator and the denominator to get:

$\sqrt{\frac{3^2}{1}}$

Simplifying the Expression

We can simplify the expression further by taking the square root of the numerator and the denominator. The square root of 3^2 is 3, and the square root of 1 is 1. Therefore, the simplified expression is:

$\boxed{3}$

Conclusion

In this article, we simplified the numerical square root expression: $\sqrt{\frac{576}{64}}$. We started by finding the prime factorization of both the numerator and the denominator, and then canceled out the common factors to simplify the expression. The final simplified expression is $\boxed{3}$. This problem demonstrates the importance of prime factorization in simplifying numerical expressions and solving complex problems in mathematics.

Discussion

The problem of simplifying the numerical square root expression: $\sqrt{\frac{576}{64}}$ is a classic example of how prime factorization can be used to simplify complex expressions. The prime factorization of the numerator and the denominator allows us to cancel out common factors and simplify the expression. This problem is an essential skill for students to learn in mathematics, as it helps them to solve complex problems efficiently and accurately.

Related Problems

• Simplify the numerical square root expression: $\sqrt{\frac{324}{81}}$
• Simplify the numerical square root expression: $\sqrt{\frac{900}{225}}$
• Simplify the numerical square root expression: $\sqrt{\frac{196}{49}}$

References

• [1] "Prime Factorization" by Math Open Reference. Retrieved from https://www.mathopenref.com/primefactorization.html
• [2] "Simplifying Square Roots" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f-simplifying-square-roots

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources.

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Introduction

In our previous article, we simplified the numerical square root expression: $\sqrt{\frac{576}{64}}$. We broke down the problem into smaller steps, starting with the prime factorization of the numerator and denominator. In this article, we will answer some frequently asked questions related to the problem.

Q&A

Q: What is the prime factorization of 576?

A: The prime factorization of 576 is 2^6 * 3^2.

Q: What is the prime factorization of 64?

A: The prime factorization of 64 is 2^6.

Q: How do I simplify the expression $\sqrt{\frac{576}{64}}$?

A: To simplify the expression, you need to find the prime factorization of both the numerator and the denominator. Then, cancel out the common factors to get:

$\sqrt{\frac{2^6 * 3^2}{2^6}}$

Q: What is the simplified expression?

A: The simplified expression is $\boxed{3}$.

Q: Why do I need to cancel out the common factors?

A: You need to cancel out the common factors to simplify the expression. In this case, both the numerator and the denominator have a factor of 2^6. Canceling out this common factor gives us:

$\sqrt{\frac{3^2}{1}}$

Q: What is the final simplified expression?

A: The final simplified expression is $\boxed{3}$.

Q: Can I use a calculator to simplify the expression?

A: Yes, you can use a calculator to simplify the expression. However, it's always a good idea to understand the underlying math and simplify the expression manually.

Q: What is the importance of prime factorization in simplifying numerical expressions?

A: Prime factorization is an essential skill in mathematics that helps you simplify complex expressions. By finding the prime factorization of a number, you can cancel out common factors and simplify the expression.

Q: Can I apply this method to other numerical expressions?

A: Yes, you can apply this method to other numerical expressions. The key is to find the prime factorization of both the numerator and the denominator and then cancel out the common factors.

Conclusion

In this article, we answered some frequently asked questions related to the problem of simplifying the numerical square root expression: $\sqrt{\frac{576}{64}}$. We provided step-by-step solutions and explanations to help you understand the underlying math. We hope this article has been helpful in clarifying any doubts you may have had.

Discussion

The problem of simplifying the numerical square root expression: $\sqrt{\frac{576}{64}}$ is a classic example of how prime factorization can be used to simplify complex expressions. The prime factorization of the numerator and the denominator allows us to cancel out common factors and simplify the expression. This problem is an essential skill for students to learn in mathematics, as it helps them to solve complex problems efficiently and accurately.

Related Problems

• Simplify the numerical square root expression: $\sqrt{\frac{324}{81}}$
• Simplify the numerical square root expression: $\sqrt{\frac{900}{225}}$
• Simplify the numerical square root expression: $\sqrt{\frac{196}{49}}$

References

• [1] "Prime Factorization" by Math Open Reference. Retrieved from https://www.mathopenref.com/primefactorization.html
• [2] "Simplifying Square Roots" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f-simplifying-square-roots

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources.