Evaluate 3 512 × 343 3 \sqrt{512 \times 343} 3 512 × 343 ​ .

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Introduction

In mathematics, evaluating expressions involving square roots is a fundamental concept that requires a deep understanding of the properties of square roots and their behavior under different operations. In this article, we will focus on evaluating the expression $3 \sqrt{512 \times 343}$, which involves the product of two numbers under a square root. We will break down the problem step by step, using various mathematical techniques to simplify the expression and arrive at the final answer.

Understanding the Expression

The given expression is $3 \sqrt{512 \times 343}$. To evaluate this expression, we need to first understand the properties of square roots and how they interact with multiplication. The 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.

Breaking Down the Expression

To evaluate the expression $3 \sqrt{512 \times 343}$, we can start by breaking down the product under the square root into its prime factors. This will allow us to simplify the expression and make it easier to work with.

Prime Factorization

The first step in breaking down the product under the square root is to find the prime factors of 512 and 343. We can start by finding the prime factorization of each number.

• 512 can be written as $2^9$.
• 343 can be written as $7^3$.

Simplifying the Expression

Now that we have the prime factorization of both numbers, we can simplify the expression by combining the prime factors under the square root.

$$3 \sqrt{512 \times 343} = 3 \sqrt{2^9 \times 7^3}$$

Using the Property of Square Roots

One of the key properties of square roots is that they can be simplified by combining the prime factors under the square root. This property states that the square root of a product is equal to the product of the square roots.

$$\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$$

Using this property, we can simplify the expression as follows:

$$3 \sqrt{2^9 \times 7^3} = 3 \times \sqrt{2^9} \times \sqrt{7^3}$$

Evaluating the Square Roots

Now that we have simplified the expression, we can evaluate the square roots of the prime factors.

• The square root of $2^9$ is $2^4 \times \sqrt{2}$.
• The square root of $7^3$ is $7^1 \times \sqrt{7}$.

Substituting the Values

We can now substitute the values of the square roots back into the expression.

$$3 \times 2^4 \times \sqrt{2} \times 7^1 \times \sqrt{7}$$

Simplifying the Expression

Now that we have substituted the values of the square roots, we can simplify the expression by combining the like terms.

$$3 \times 2^4 \times 7^1 \times \sqrt{2} \times \sqrt{7}$$

Evaluating the Expression

Finally, we can evaluate the expression by multiplying the numbers together.

$$3 \times 2^4 \times 7^1 \times \sqrt{2} \times \sqrt{7} = 3 \times 16 \times 7 \times \sqrt{14}$$

Final Answer

The final answer to the expression $3 \sqrt{512 \times 343}$ is $3 \times 16 \times 7 \times \sqrt{14} = 336 \times \sqrt{14}$.

Conclusion

In this article, we evaluated the expression $3 \sqrt{512 \times 343}$ by breaking down the product under the square root into its prime factors, using the property of square roots to simplify the expression, and finally evaluating the expression by multiplying the numbers together. The final answer to the expression is $336 \times \sqrt{14}$.

Frequently Asked Questions

• What is the property of square roots that allows us to simplify the expression? The property of square roots states that the square root of a product is equal to the product of the square roots.
• How do we evaluate the square roots of the prime factors? We evaluate the square roots of the prime factors by taking the square root of each prime factor individually.
• What is the final answer to the expression $3 \sqrt{512 \times 343}$? The final answer to the expression $3 \sqrt{512 \times 343}$ is $336 \times \sqrt{14}$.

References

• [1] Khan Academy. (n.d.). Square Roots. Retrieved from https://www.khanacademy.org/math/algebra/x2f2f7c/square-roots
• [2] Math Open Reference. (n.d.). Square Root. Retrieved from https://www.mathopenref.com/sqrt.html

Related Articles

• Evaluating Expressions Involving Square Roots
• Properties of Square Roots
• Simplifying Expressions Involving Square Roots
  

Introduction

Evaluating expressions involving square roots can be a challenging task, especially when dealing with complex numbers and variables. In this article, we will address some of the most frequently asked questions related to evaluating expressions involving square roots.

Q1: What is the property of square roots that allows us to simplify the expression?

A1: The property of square roots states that the square root of a product is equal to the product of the square roots. This property can be expressed as:

$$\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$$

Q2: How do we evaluate the square roots of the prime factors?

A2: To evaluate the square roots of the prime factors, we take the square root of each prime factor individually. For example, if we have the expression $\sqrt{2^9 \times 7^3}$, we can evaluate the square roots of the prime factors as follows:

• The square root of $2^9$ is $2^4 \times \sqrt{2}$.
• The square root of $7^3$ is $7^1 \times \sqrt{7}$.

Q3: What is the final answer to the expression $3 \sqrt{512 \times 343}$?

A3: The final answer to the expression $3 \sqrt{512 \times 343}$ is $336 \times \sqrt{14}$.

Q4: How do we simplify expressions involving square roots?

A4: To simplify expressions involving square roots, we can use the property of square roots to combine the prime factors under the square root. This can be done by multiplying the square roots of the prime factors together.

Q5: What are some common mistakes to avoid when evaluating expressions involving square roots?

A5: Some common mistakes to avoid when evaluating expressions involving square roots include:

• Not using the property of square roots to simplify the expression.
• Not evaluating the square roots of the prime factors correctly.
• Not combining the like terms correctly.

Q6: How do we evaluate expressions involving square roots with variables?

A6: To evaluate expressions involving square roots with variables, we can use the same techniques as before, but we need to be careful when dealing with variables. We need to make sure that we are using the correct property of square roots and that we are evaluating the square roots of the variables correctly.

Q7: What are some real-world applications of evaluating expressions involving square roots?

A7: Some real-world applications of evaluating expressions involving square roots include:

• Calculating the area and perimeter of a square or rectangle.
• Calculating the volume of a cube or rectangular prism.
• Calculating the distance between two points on a coordinate plane.

Q8: How do we use technology to evaluate expressions involving square roots?

A8: We can use technology such as calculators or computer software to evaluate expressions involving square roots. These tools can help us to simplify the expression and evaluate the square roots of the prime factors.

Q9: What are some tips for evaluating expressions involving square roots?

A9: Some tips for evaluating expressions involving square roots include:

• Make sure to use the property of square roots to simplify the expression.
• Evaluate the square roots of the prime factors correctly.
• Combine the like terms correctly.

Q10: How do we check our work when evaluating expressions involving square roots?

A10: To check our work when evaluating expressions involving square roots, we can use the following steps:

• Simplify the expression using the property of square roots.
• Evaluate the square roots of the prime factors.
• Combine the like terms.
• Check the final answer to make sure it is correct.

Conclusion

Evaluating expressions involving square roots can be a challenging task, but by using the property of square roots and following the correct steps, we can simplify the expression and arrive at the final answer. By following the tips and techniques outlined in this article, we can become more confident and proficient in evaluating expressions involving square roots.

Frequently Asked Questions

• What is the property of square roots that allows us to simplify the expression?
• How do we evaluate the square roots of the prime factors?
• What is the final answer to the expression $3 \sqrt{512 \times 343}$?
• How do we simplify expressions involving square roots?
• What are some common mistakes to avoid when evaluating expressions involving square roots?
• How do we evaluate expressions involving square roots with variables?
• What are some real-world applications of evaluating expressions involving square roots?
• How do we use technology to evaluate expressions involving square roots?
• What are some tips for evaluating expressions involving square roots?
• How do we check our work when evaluating expressions involving square roots?

References

• [1] Khan Academy. (n.d.). Square Roots. Retrieved from https://www.khanacademy.org/math/algebra/x2f2f7c/square-roots
• [2] Math Open Reference. (n.d.). Square Root. Retrieved from https://www.mathopenref.com/sqrt.html

Related Articles

• Evaluating Expressions Involving Square Roots
• Properties of Square Roots
• Simplifying Expressions Involving Square Roots