Simplify The Expression: 12 245 12 \sqrt{245} 12 245 ​

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Introduction

Simplifying expressions involving square roots is a crucial skill in mathematics, particularly in algebra and geometry. In this article, we will focus on simplifying the expression $12 \sqrt{245}$, which involves finding the prime factorization of the number under the square root sign and then simplifying the expression accordingly.

Understanding Square Roots

Before we dive into simplifying the expression, let's briefly review what square roots are. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16. In mathematical notation, we write this as $\sqrt{16} = 4$.

Prime Factorization

To simplify the expression $12 \sqrt{245}$, we need to find the prime factorization of the number under the square root sign, which is 245. Prime factorization involves breaking down a number into its prime factors, which are numbers that can only be divided by 1 and themselves.

Finding the Prime Factorization of 245

To find the prime factorization of 245, we can start by dividing it by the smallest prime number, which is 2. However, 245 is an odd number, so we can start by dividing it by 3, which is the next smallest prime number.

245 ÷ 5 = 49
49 ÷ 7 = 7
7 ÷ 1 = 7

So, the prime factorization of 245 is $5 \times 7 \times 7$.

Simplifying the Expression

Now that we have the prime factorization of 245, we can simplify the expression $12 \sqrt{245}$. We can rewrite the expression as $12 \sqrt{5 \times 7 \times 7}$.

Breaking Down the Expression

We can break down the expression further by taking the square root of each prime factor. The square root of 5 is $\sqrt{5}$, the square root of 7 is $\sqrt{7}$, and the square root of 7 is also $\sqrt{7}$.

Simplifying the Expression Further

Now that we have broken down the expression, we can simplify it further by combining the square roots. We can rewrite the expression as $12 \times \sqrt{5} \times \sqrt{7} \times \sqrt{7}$.

Canceling Out the Square Roots

We can cancel out the square roots of 7 by multiplying them together, which gives us $\sqrt{7} \times \sqrt{7} = 7$. So, the expression simplifies to $12 \times \sqrt{5} \times 7$.

Final Simplification

Now that we have simplified the expression, we can multiply the numbers together to get the final result. $12 \times 7 = 84$, so the final simplified expression is $84 \sqrt{5}$.

Conclusion

In this article, we simplified the expression $12 \sqrt{245}$ by finding the prime factorization of the number under the square root sign and then simplifying the expression accordingly. We broke down the expression into its prime factors, took the square root of each factor, and then combined the square roots to simplify the expression further. The final simplified expression is $84 \sqrt{5}$.

Frequently Asked Questions

• What is the prime factorization of 245? The prime factorization of 245 is $5 \times 7 \times 7$.
• How do you simplify an expression involving a square root? To simplify an expression involving a square root, you need to find the prime factorization of the number under the square root sign and then simplify the expression accordingly.
• What is the final simplified expression for $12 \sqrt{245}$? The final simplified expression for $12 \sqrt{245}$ is $84 \sqrt{5}$.

Further Reading

If you want to learn more about simplifying expressions involving square roots, you can check out the following resources:

• Khan Academy: Simplifying Square Roots
• Mathway: Simplifying Square Roots
• Wolfram Alpha: Simplifying Square Roots

References

• "Algebra" by Michael Artin
• "Geometry" by Michael Spivak
• "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer
  

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Introduction

In our previous article, we simplified the expression $12 \sqrt{245}$ by finding the prime factorization of the number under the square root sign and then simplifying the expression accordingly. In this article, we will answer some frequently asked questions related to simplifying expressions involving square roots.

Q&A

Q: What is the prime factorization of 245?

A: The prime factorization of 245 is $5 \times 7 \times 7$.

Q: How do you simplify an expression involving a square root?

A: To simplify an expression involving a square root, you need to find the prime factorization of the number under the square root sign and then simplify the expression accordingly.

Q: What is the final simplified expression for $12 \sqrt{245}$?

A: The final simplified expression for $12 \sqrt{245}$ is $84 \sqrt{5}$.

Q: Can you explain the concept of prime factorization?

A: Prime factorization is the process of breaking down a number into its prime factors, which are numbers that can only be divided by 1 and themselves.

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 even, you can continue dividing it by 2 until it is no longer even. Then, you can move on to the next prime number, which is 3, and continue dividing the number by 3 until it is no longer divisible by 3. You can continue this process until you have found all the prime factors of the number.

Q: Can you give an example of prime factorization?

A: Yes, let's consider the number 12. The prime factorization of 12 is $2 \times 2 \times 3$, because 12 can be divided by 2 twice, and then by 3.

Q: How do you simplify an expression involving a square root and a number?

A: To simplify an expression involving a square root and a number, you can multiply the number by the square root. For example, if you have the expression $3 \sqrt{4}$, you can simplify it by multiplying 3 by the square root of 4, which is 2.

Q: Can you explain the concept of simplifying expressions involving square roots?

A: Simplifying expressions involving square roots involves finding the prime factorization of the number under the square root sign and then simplifying the expression accordingly. This can involve canceling out square roots, multiplying numbers together, and rearranging the expression to make it easier to read.

Q: How do you know when an expression is simplified?

A: An expression is simplified when it can no longer be simplified further. This means that there are no more square roots that can be canceled out, and no more numbers that can be multiplied together.

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions involving square roots. We explained the concept of prime factorization, how to find the prime factorization of a number, and how to simplify expressions involving square roots. We also provided examples and explanations to help illustrate the concepts.

Frequently Asked Questions

• What is the prime factorization of 245?
• How do you simplify an expression involving a square root?
• What is the final simplified expression for $12 \sqrt{245}$?
• Can you explain the concept of prime factorization?
• How do you find the prime factorization of a number?
• Can you give an example of prime factorization?
• How do you simplify an expression involving a square root and a number?
• Can you explain the concept of simplifying expressions involving square roots?
• How do you know when an expression is simplified?

Further Reading

If you want to learn more about simplifying expressions involving square roots, you can check out the following resources:

• Khan Academy: Simplifying Square Roots
• Mathway: Simplifying Square Roots
• Wolfram Alpha: Simplifying Square Roots

References

• "Algebra" by Michael Artin
• "Geometry" by Michael Spivak
• "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer