A Science Teacher Told His Students That The Expression 4 M 4 144 N 4 \frac{4 M^4}{144 N^4} 144 N 4 4 M 4 ​ Represents The Area In Square Units Of The Contaminated Square Region In Their Activity. What Is The Width Of The Region In Simplest Form?A. $\frac{2 M^2}{12

Understanding the Given Expression

The science teacher has provided an expression $\frac{4 m^4}{144 n^4}$, which represents the area in square units of the contaminated square region in their activity. To find the width of the region, we need to simplify the given expression and identify the value of the width.

Breaking Down the Expression

The given expression can be broken down into two parts: the numerator and the denominator. The numerator is $4 m^4$, and the denominator is $144 n^4$. To simplify the expression, we need to factorize both the numerator and the denominator.

Factorizing the Numerator

The numerator $4 m^4$ can be factorized as $4 \times m^4$. Since $4$ is a constant, we can rewrite it as $2^2$. Therefore, the numerator can be rewritten as $2^2 \times m^4$.

Factorizing the Denominator

The denominator $144 n^4$ can be factorized as $144 \times n^4$. Since $144$ is a constant, we can rewrite it as $2^4 \times 3^2$. Therefore, the denominator can be rewritten as $2^4 \times 3^2 \times n^4$.

Simplifying the Expression

Now that we have factorized both the numerator and the denominator, we can simplify the expression by canceling out common factors. The numerator has a factor of $2^2$, and the denominator has a factor of $2^4$. We can cancel out two factors of $2$ from the numerator and the denominator, leaving us with $2^2 \times m^4$ in the numerator and $2^2 \times 3^2 \times n^4$ in the denominator.

Canceling Out Common Factors

After canceling out the common factors, the expression becomes $\frac{2^2 \times m^4}{2^2 \times 3^2 \times n^4}$. We can simplify this expression further by canceling out the common factor of $2^2$ from the numerator and the denominator.

Simplifying the Expression Further

After canceling out the common factor of $2^2$, the expression becomes $\frac{m^4}{3^2 \times n^4}$. We can simplify this expression further by rewriting $3^2$ as $9$.

Rewriting the Expression

The expression can be rewritten as $\frac{m^4}{9 \times n^4}$. To simplify this expression further, we need to find the value of the width of the contaminated region.

Finding the Width of the Region

The width of the region is represented by the value of $m$. To find the value of $m$, we need to simplify the expression further.

Simplifying the Expression to Find the Width

The expression $\frac{m^4}{9 \times n^4}$ can be simplified by rewriting $m^4$ as $m^2 \times m^2$. Therefore, the expression becomes $\frac{m^2 \times m^2}{9 \times n^4}$.

Finding the Value of the Width

To find the value of the width, we need to simplify the expression further. We can rewrite the expression as $\frac{m^2}{3 \times n^2} \times \frac{m^2}{3 \times n^2}$.

Simplifying the Expression to Find the Value of the Width

The expression $\frac{m^2}{3 \times n^2} \times \frac{m^2}{3 \times n^2}$ can be simplified by canceling out the common factor of $3 \times n^2$ from the numerator and the denominator.

Finding the Value of the Width

After canceling out the common factor of $3 \times n^2$, the expression becomes $\frac{m^2}{3 \times n^2} \times \frac{m^2}{3 \times n^2}$. We can simplify this expression further by rewriting it as $\left(\frac{m^2}{3 \times n^2}\right)^2$.

Simplifying the Expression to Find the Value of the Width

The expression $\left(\frac{m^2}{3 \times n^2}\right)^2$ can be simplified by rewriting it as $\frac{m^4}{9 \times n^4}$.

Finding the Value of the Width

To find the value of the width, we need to simplify the expression further. We can rewrite the expression as $\frac{m^2}{3 \times n^2}$.

Simplifying the Expression to Find the Value of the Width

The expression $\frac{m^2}{3 \times n^2}$ represents the width of the contaminated region. Therefore, the width of the region is $\frac{2 m^2}{12 n^2}$.

Conclusion

In conclusion, the width of the contaminated region is $\frac{2 m^2}{12 n^2}$. This is the simplest form of the expression, and it represents the width of the region in square units.

Final Answer

The final answer is $\boxed{\frac{2 m^2}{12 n^2}}$.

Q: What is the expression that represents the area of the contaminated square region?

A: The expression that represents the area of the contaminated square region is $\frac{4 m^4}{144 n^4}$.

Q: How do I simplify the expression to find the width of the region?

A: To simplify the expression, you need to factorize both the numerator and the denominator, cancel out common factors, and rewrite the expression in its simplest form.

Q: What is the value of the width of the contaminated region?

A: The value of the width of the contaminated region is $\frac{2 m^2}{12 n^2}$.

Q: Why do I need to cancel out common factors to simplify the expression?

A: Canceling out common factors helps to simplify the expression and make it easier to understand. It also helps to identify the value of the width of the region.

Q: How do I rewrite the expression in its simplest form?

A: To rewrite the expression in its simplest form, you need to factorize both the numerator and the denominator, cancel out common factors, and rewrite the expression using the simplest form of the variables.

Q: What is the significance of the width of the contaminated region?

A: The width of the contaminated region is an important factor in understanding the extent of the contamination. It helps to identify the area that needs to be cleaned or treated.

Q: Can I use the expression to find the area of the contaminated region?

A: Yes, you can use the expression to find the area of the contaminated region. However, you need to simplify the expression first to find the value of the width of the region.

Q: How do I apply the concept of simplifying expressions to real-world problems?

A: You can apply the concept of simplifying expressions to real-world problems by identifying the variables and constants in the expression, factorizing both the numerator and the denominator, canceling out common factors, and rewriting the expression in its simplest form.

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

A: Some common mistakes to avoid when simplifying expressions include:

• Not factorizing both the numerator and the denominator
• Not canceling out common factors
• Not rewriting the expression in its simplest form
• Not identifying the value of the width of the region

Q: How do I check my work when simplifying expressions?

A: To check your work when simplifying expressions, you need to:

• Verify that you have factorized both the numerator and the denominator
• Verify that you have canceled out common factors
• Verify that you have rewritten the expression in its simplest form
• Verify that you have identified the value of the width of the region

Q: Can I use technology to simplify expressions?

A: Yes, you can use technology to simplify expressions. Many calculators and computer software programs have built-in functions that can simplify expressions and identify the value of the width of the region.

Q: How do I apply the concept of simplifying expressions to different types of problems?

A: You can apply the concept of simplifying expressions to different types of problems by identifying the variables and constants in the expression, factorizing both the numerator and the denominator, canceling out common factors, and rewriting the expression in its simplest form.

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

A: Some real-world applications of simplifying expressions include:

• Calculating the area of a rectangle or a square
• Finding the perimeter of a rectangle or a square
• Identifying the value of the width of a contaminated region
• Calculating the volume of a rectangular prism or a cube

Q: How do I communicate the results of simplifying expressions to others?

A: To communicate the results of simplifying expressions to others, you need to:

• Clearly explain the steps you took to simplify the expression
• Identify the value of the width of the region
• Provide examples or illustrations to help others understand the concept
• Use clear and concise language to explain the results