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 $\square$

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Understanding the Problem

To simplify the given expression, we need to start by finding the prime factorization of both the numerator and the denominator. The prime factorization of a number is the expression of that number as the product of its prime factors.

Prime Factorization of 576

The prime factorization of 576 is a crucial step in simplifying the given expression. To find the prime factorization of 576, we need to break it down into its prime factors.

576 can be divided by 2, which is a prime number. So, we can write 576 as:

576 = 2 × 288

288 can also be divided by 2, so we can write it as:

288 = 2 × 144

144 can be divided by 2 again, so we can write it as:

144 = 2 × 72

72 can be divided by 2, so we can write it as:

72 = 2 × 36

36 can be divided by 2, so we can write it as:

36 = 2 × 18

18 can be divided by 2, so we can write it as:

18 = 2 × 9

9 can be divided by 3, which is a prime number, so we can write it as:

9 = 3 × 3

Now, we can combine all the factors to get the prime factorization of 576:

576 = 2^6 × 3^2

Prime Factorization of 64

The prime factorization of 64 is also a crucial step in simplifying the given expression. To find the prime factorization of 64, we need to break it down into its prime factors.

64 can be divided by 2, which is a prime number, so we can write 64 as:

64 = 2 × 32

32 can be divided by 2, so we can write it as:

32 = 2 × 16

16 can be divided by 2, so we can write it as:

16 = 2 × 8

8 can be divided by 2, so we can write it as:

8 = 2 × 4

4 can be divided by 2, so we can write it as:

4 = 2 × 2

Now, we can combine all the factors to get the prime factorization of 64:

64 = 2^6

Simplifying the Expression

Now that we have the prime factorization of both the numerator and the denominator, we can simplify the expression.

The expression 57664\sqrt{\frac{576}{64}} can be simplified by canceling out the common factors in the numerator and the denominator.

The numerator has a factor of 2^6, and the denominator has a factor of 2^6. We can cancel out these common factors to get:

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

Now, we can simplify the expression by canceling out the common factors:

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

The square root of 1 is 1, so we can simplify the expression further:

32\sqrt{3^2}

The square root of 3^2 is 3, so the final simplified form of the expression is:

3\boxed{3}

Conclusion

In this article, we simplified the expression 57664\sqrt{\frac{576}{64}} by finding the prime factorization of both the numerator and the denominator. We canceled out the common factors in the numerator and the denominator to simplify the expression. The final simplified form of the expression 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 I simplify the expression 57664\sqrt{\frac{576}{64}}?

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

Q: What are the common factors in the numerator and the denominator?

A: The common factors in the numerator and the denominator are 2^6.

Q: How do I cancel out the common factors?

A: To cancel out the common factors, you need to divide both the numerator and the denominator by the common factors. In this case, you need to divide both the numerator and the denominator by 2^6.

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\boxed{3}.

Q: Why is the simplified form of the expression 3\boxed{3}?

A: The simplified form of the expression 3\boxed{3} because the square root of 3^2 is 3.

Q: What is the square root of 3^2?

A: The square root of 3^2 is 3.

Q: Can I simplify the expression 57664\sqrt{\frac{576}{64}} further?

A: No, the expression 57664\sqrt{\frac{576}{64}} cannot be simplified further.

Q: Why can't the expression 57664\sqrt{\frac{576}{64}} be simplified further?

A: The expression 57664\sqrt{\frac{576}{64}} cannot be simplified further because there are no more common factors to cancel out.

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\boxed{3}.

Additional Tips and Tricks

  • When simplifying expressions, always look for common factors to cancel out.
  • Use prime factorization to find the common factors in the numerator and the denominator.
  • Cancel out the common factors by dividing both the numerator and the denominator by the common factors.
  • Simplify the expression by canceling out the common factors.
  • Check if the expression can be simplified further by looking for any remaining common factors.

Conclusion

In this article, we answered some frequently asked questions about simplifying the expression 57664\sqrt{\frac{576}{64}}. We provided step-by-step instructions on how to simplify the expression and explained why the simplified form is 3\boxed{3}. We also provided additional tips and tricks on how to simplify expressions.