Simplify:$\[ 7 \sqrt{32} - 6 \sqrt{72} \\]Enter Your Answer, In Simplest Radical Form, In The Box.$\[ \square \\]

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Understanding the Problem

When dealing with expressions involving square roots, it's essential to simplify them to their most basic form. This involves breaking down the square roots into their prime factors and then simplifying the expression. In this problem, we're given the expression $7 \sqrt{32} - 6 \sqrt{72}$, and we need to simplify it to its simplest radical form.

Breaking Down the Square Roots

To simplify the expression, we need to break down the square roots into their prime factors. We can start by finding the prime factors of 32 and 72.

Prime Factors of 32

The prime factors of 32 are 2 and 2 and 2 and 2 and 2 and 2, which can be written as $2^5$. This means that $\sqrt{32}$ can be simplified as $\sqrt{2^5}$.

Prime Factors of 72

The prime factors of 72 are 2 and 2 and 2 and 3 and 3, which can be written as $2^3 \times 3^2$. This means that $\sqrt{72}$ can be simplified as $\sqrt{2^3 \times 3^2}$.

Simplifying the Expression

Now that we have broken down the square roots into their prime factors, we can simplify the expression.

Simplifying $7 \sqrt{32}$

We can simplify $7 \sqrt{32}$ as $7 \times \sqrt{2^5}$, which is equal to $7 \times 2^2 \times \sqrt{2}$, or $28 \sqrt{2}$.

Simplifying $6 \sqrt{72}$

We can simplify $6 \sqrt{72}$ as $6 \times \sqrt{2^3 \times 3^2}$, which is equal to $6 \times 2 \times 3 \times \sqrt{2}$, or $36 \sqrt{2}$.

Combining the Simplified Expressions

Now that we have simplified both expressions, we can combine them to get the final result.

Final Result

The final result is $28 \sqrt{2} - 36 \sqrt{2}$, which can be simplified as $-8 \sqrt{2}$.

Conclusion

In this problem, we simplified the expression $7 \sqrt{32} - 6 \sqrt{72}$ to its simplest radical form. We broke down the square roots into their prime factors and then simplified the expression. The final result is $-8 \sqrt{2}$.

Tips and Tricks

When dealing with expressions involving square roots, it's essential to break down the square roots into their prime factors and then simplify the expression. This involves finding the prime factors of the numbers inside the square roots and then simplifying the expression.

Common Mistakes to Avoid

One common mistake to avoid is not breaking down the square roots into their prime factors. This can lead to a more complicated expression that is difficult to simplify.

Best Practices

The best practice when dealing with expressions involving square roots is to break down the square roots into their prime factors and then simplify the expression. This involves finding the prime factors of the numbers inside the square roots and then simplifying the expression.

Real-World Applications

Simplifying expressions involving square roots has many real-world applications. For example, in physics, we often encounter expressions involving square roots when dealing with problems involving motion and energy.

Example 1

Suppose we have a particle moving in a straight line with a velocity of $v = 2 \sqrt{3}$ m/s. We want to find the kinetic energy of the particle. The kinetic energy is given by the formula $E_k = \frac{1}{2} m v^2$, where $m$ is the mass of the particle.

Example 2

Suppose we have a spring with a spring constant of $k = 2 \sqrt{2}$ N/m. We want to find the potential energy of the spring when it is stretched by a distance of $x = 3$ m. The potential energy is given by the formula $E_p = \frac{1}{2} k x^2$.

Conclusion

In this article, we simplified the expression $7 \sqrt{32} - 6 \sqrt{72}$ to its simplest radical form. We broke down the square roots into their prime factors and then simplified the expression. The final result is $-8 \sqrt{2}$. We also discussed the importance of breaking down square roots into their prime factors and provided tips and tricks for simplifying expressions involving square roots.

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Frequently Asked Questions

Q: What is the prime factorization of 32?

A: The prime factorization of 32 is $2^5$.

Q: What is the prime factorization of 72?

A: The prime factorization of 72 is $2^3 \times 3^2$.

Q: How do I simplify $7 \sqrt{32}$?

A: To simplify $7 \sqrt{32}$, we can break down the square root into its prime factors. Since the prime factorization of 32 is $2^5$, we can simplify $7 \sqrt{32}$ as $7 \times 2^2 \times \sqrt{2}$, or $28 \sqrt{2}$.

Q: How do I simplify $6 \sqrt{72}$?

A: To simplify $6 \sqrt{72}$, we can break down the square root into its prime factors. Since the prime factorization of 72 is $2^3 \times 3^2$, we can simplify $6 \sqrt{72}$ as $6 \times 2 \times 3 \times \sqrt{2}$, or $36 \sqrt{2}$.

Q: What is the final result of the expression $7 \sqrt{32} - 6 \sqrt{72}$?

A: The final result of the expression $7 \sqrt{32} - 6 \sqrt{72}$ is $-8 \sqrt{2}$.

Q: Why is it important to break down square roots into their prime factors?

A: Breaking down square roots into their prime factors is essential when simplifying expressions involving square roots. It allows us to identify the common factors and simplify the expression.

Q: What are some common mistakes to avoid when simplifying expressions involving square roots?

A: Some common mistakes to avoid when simplifying expressions involving square roots include:

• Not breaking down the square roots into their prime factors
• Not identifying the common factors
• Not simplifying the expression correctly

Q: What are some real-world applications of simplifying expressions involving square roots?

A: Simplifying expressions involving square roots has many real-world applications, including:

• Physics: When dealing with problems involving motion and energy
• Engineering: When designing and building structures
• Computer Science: When working with algorithms and data structures

Q: How can I practice simplifying expressions involving square roots?

A: You can practice simplifying expressions involving square roots by:

• Working through examples and exercises
• Using online resources and tools
• Practicing with real-world problems and applications

Conclusion

In this Q&A article, we answered some of the most frequently asked questions about simplifying expressions involving square roots. We covered topics such as prime factorization, simplifying expressions, and real-world applications. We also provided tips and tricks for avoiding common mistakes and practicing simplifying expressions involving square roots.