Simplify: $\sqrt{\frac{576}{64}}$The Prime Factorization Of 576 Is $\square$.The Prime Factorization Of 64 Is $\square$.The Expression $\sqrt{\frac{576}{64}}$ In Simplest Form Is $\square$.

by ADMIN 196 views

Prime Factorization of 576 and 64

To simplify the expression 57664\sqrt{\frac{576}{64}}, we need to find the prime factorization of both 576 and 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

The prime factorization of 576 is the expression of 576 as the product of its prime factors. To find the prime factorization of 576, we can start by dividing it by the smallest prime number, which is 2.

576 ÷ 2 = 288 288 ÷ 2 = 144 144 ÷ 2 = 72 72 ÷ 2 = 36 36 ÷ 2 = 18 18 ÷ 2 = 9 9 ÷ 3 = 3

Therefore, the prime factorization of 576 is:

576 = 2^6 × 3^2

The Prime Factorization of 64

The prime factorization of 64 is the expression of 64 as the product of its prime factors. To find the prime factorization of 64, we can start by dividing it by the smallest prime number, which is 2.

64 ÷ 2 = 32 32 ÷ 2 = 16 16 ÷ 2 = 8 8 ÷ 2 = 4 4 ÷ 2 = 2 2 ÷ 2 = 1

Therefore, the prime factorization of 64 is:

64 = 2^6

Simplifying the Expression

Now that we have the prime factorization of both 576 and 64, we can simplify the expression 57664\sqrt{\frac{576}{64}}.

57664\sqrt{\frac{576}{64}} = 26×3226\sqrt{\frac{2^6 × 3^2}{2^6}}

We can simplify the expression by canceling out the common factors in the numerator and denominator.

26×3226\sqrt{\frac{2^6 × 3^2}{2^6}} = 32\sqrt{3^2}

32\sqrt{3^2} = 3

Therefore, the expression 57664\sqrt{\frac{576}{64}} in simplest form is 3.

Conclusion

In this article, we simplified the expression 57664\sqrt{\frac{576}{64}} by finding the prime factorization of both 576 and 64. We then used the prime factorization to simplify the expression and found that the expression in simplest form is 3.

Frequently Asked Questions

  • What is the prime factorization of 576?
    • The prime factorization of 576 is 2^6 × 3^2.
  • What is the prime factorization of 64?
    • The prime factorization of 64 is 2^6.
  • How do you simplify the expression 57664\sqrt{\frac{576}{64}}?
    • To simplify the expression, you need to find the prime factorization of both 576 and 64, and then cancel out the common factors in the numerator and denominator.

Final Answer

The final answer is: 3\boxed{3}

Frequently Asked Questions

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 you simplify the expression 57664\sqrt{\frac{576}{64}}?

A: To simplify the expression, you need to find the prime factorization of both 576 and 64, and then cancel out the common factors in the numerator and denominator.

Q: What is the simplified form of the expression 57664\sqrt{\frac{576}{64}}?

A: The simplified form of the expression 57664\sqrt{\frac{576}{64}} is 3.

Q: Can you explain the steps to simplify the expression 57664\sqrt{\frac{576}{64}}?

A: Here are the steps to simplify the expression:

  1. Find the prime factorization of both 576 and 64.
  2. Cancel out the common factors in the numerator and denominator.
  3. Simplify the expression to its final form.

Q: What is the final answer to the expression 57664\sqrt{\frac{576}{64}}?

A: The final answer to the expression 57664\sqrt{\frac{576}{64}} is 3.

Q: Can you provide more examples of simplifying expressions with square roots?

A: Here are a few more examples:

  • 14416\sqrt{\frac{144}{16}} = 3\boxed{3}
  • 900100\sqrt{\frac{900}{100}} = 3\boxed{3}
  • 32436\sqrt{\frac{324}{36}} = 3\boxed{3}

Q: How do you know when to simplify an expression with a square root?

A: You should simplify an expression with a square root when the numerator and denominator have common factors that can be canceled out.

Q: Can you explain the concept of prime factorization?

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

Q: How do you find the prime factorization of a number?

A: To find the prime factorization of a number, you can start by dividing it by the smallest prime number, which is 2. If the number is divisible by 2, then you can continue dividing it by 2 until it is no longer divisible. Then, you can move on to the next prime number, which is 3, and repeat the process.

Conclusion

In this article, we provided answers to frequently asked questions about simplifying the expression 57664\sqrt{\frac{576}{64}}. We explained the steps to simplify the expression, provided examples of simplifying expressions with square roots, and discussed the concept of prime factorization.

Final Answer

The final answer is: 3\boxed{3}