Which Expression Will Simplify To 1? Assume $m \neq 9$ And $m \neq -9$.A. $\left(\frac{m+9}{m-9}\right)\left(\frac{m+9}{m-9}\right$\] B. $\left(\frac{m+9}{m-9}\right)\left(\frac{m-9}{m+9}\right$\] C.

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently. When dealing with fractions, we often need to simplify them to their simplest form. In this article, we will explore which expression will simplify to 1, given certain conditions.

Understanding the Conditions

Before we dive into the expressions, let's understand the conditions given in the problem. We are told that $m \neq 9$ and $m \neq -9$. This means that the variable $m$ cannot be equal to 9 or -9. These conditions are crucial in determining which expression will simplify to 1.

Expression A

Let's start by analyzing Expression A:

$\left(\frac{m+9}{m-9}\right)\left(\frac{m+9}{m-9}\right)$

To simplify this expression, we need to multiply the two fractions together. When multiplying fractions, we multiply the numerators together and the denominators together.

$\left(\frac{m+9}{m-9}\right)\left(\frac{m+9}{m-9}\right) = \frac{(m+9)(m+9)}{(m-9)(m-9)}$

Now, let's simplify the numerator and the denominator separately.

Numerator: $(m+9)(m+9) = m^2 + 18m + 81$

Denominator: $(m-9)(m-9) = m^2 - 18m + 81$

As we can see, the numerator and the denominator are identical. This means that the expression will simplify to 1, but only if $m \neq 9$ and $m \neq -9$. If $m$ is equal to 9 or -9, the expression will not simplify to 1.

Expression B

Now, let's analyze Expression B:

$\left(\frac{m+9}{m-9}\right)\left(\frac{m-9}{m+9}\right)$

To simplify this expression, we need to multiply the two fractions together. When multiplying fractions, we multiply the numerators together and the denominators together.

$\left(\frac{m+9}{m-9}\right)\left(\frac{m-9}{m+9}\right) = \frac{(m+9)(m-9)}{(m-9)(m+9)}$

Now, let's simplify the numerator and the denominator separately.

Numerator: $(m+9)(m-9) = m^2 - 81$

Denominator: $(m-9)(m+9) = m^2 - 81$

As we can see, the numerator and the denominator are identical. This means that the expression will simplify to 1, regardless of the value of $m$. The conditions $m \neq 9$ and $m \neq -9$ do not affect the simplification of this expression.

Conclusion

In conclusion, Expression B will simplify to 1, regardless of the value of $m$. The conditions $m \neq 9$ and $m \neq -9$ do not affect the simplification of this expression. On the other hand, Expression A will simplify to 1 only if $m \neq 9$ and $m \neq -9$. If $m$ is equal to 9 or -9, the expression will not simplify to 1.

Why Expression B Simplifies to 1

Expression B simplifies to 1 because the numerator and the denominator are identical. When we multiply the two fractions together, the numerator and the denominator cancel each other out, leaving us with 1.

$\left(\frac{m+9}{m-9}\right)\left(\frac{m-9}{m+9}\right) = \frac{(m+9)(m-9)}{(m-9)(m+9)} = \frac{m^2 - 81}{m^2 - 81} = 1$

Why Expression A Does Not Simplify to 1

Expression A does not simplify to 1 because the numerator and the denominator are not identical. When we multiply the two fractions together, the numerator and the denominator do not cancel each other out, leaving us with a different expression.

$\left(\frac{m+9}{m-9}\right)\left(\frac{m+9}{m-9}\right) = \frac{(m+9)(m+9)}{(m-9)(m-9)} = \frac{m^2 + 18m + 81}{m^2 - 18m + 81}$

Final Thoughts

In conclusion, Expression B will simplify to 1, regardless of the value of $m$. The conditions $m \neq 9$ and $m \neq -9$ do not affect the simplification of this expression. On the other hand, Expression A will simplify to 1 only if $m \neq 9$ and $m \neq -9$. If $m$ is equal to 9 or -9, the expression will not simplify to 1.

References

• [1] Khan Academy. (n.d.). Multiplying and dividing fractions. Retrieved from https://www.khanacademy.org/math/algebra/x2f6f4c7/x2f6f4c8/x2f6f4c9
• [2] Math Open Reference. (n.d.). Fractions. Retrieved from https://www.mathopenref.com/fractions.html

Additional Resources

• [1] Algebra.com. (n.d.). Simplifying fractions. Retrieved from https://www.algebra.com/algebra/homework/simplifying-fractions/simplifying-fractions.html
• [2] Purplemath. (n.d.). Fractions. Retrieved from https://www.purplemath.com/modules/fractions.htm
  Frequently Asked Questions (FAQs) =====================================

In this article, we will answer some of the most frequently asked questions related to the expressions that simplify to 1.

Q: What is the condition for Expression A to simplify to 1?

A: Expression A will simplify to 1 only if $m \neq 9$ and $m \neq -9$. If $m$ is equal to 9 or -9, the expression will not simplify to 1.

Q: Why does Expression B simplify to 1?

A: Expression B simplifies to 1 because the numerator and the denominator are identical. When we multiply the two fractions together, the numerator and the denominator cancel each other out, leaving us with 1.

Q: What is the difference between Expression A and Expression B?

A: The main difference between Expression A and Expression B is the order of the fractions. In Expression A, the two fractions are multiplied together in the same order, while in Expression B, the fractions are multiplied together in reverse order.

Q: Can Expression A simplify to 1 if $m$ is equal to 9 or -9?

A: No, Expression A will not simplify to 1 if $m$ is equal to 9 or -9. In fact, if $m$ is equal to 9 or -9, the expression will not be defined.

Q: Can Expression B simplify to 1 if $m$ is equal to 9 or -9?

A: Yes, Expression B will simplify to 1 regardless of the value of $m$. The conditions $m \neq 9$ and $m \neq -9$ do not affect the simplification of this expression.

Q: What is the significance of the conditions $m \neq 9$ and $m \neq -9$?

A: The conditions $m \neq 9$ and $m \neq -9$ are important because they prevent the expression from being undefined. If $m$ is equal to 9 or -9, the expression will not be defined, and it will not simplify to 1.

Q: Can Expression A and Expression B be used to solve real-world problems?

A: Yes, Expression A and Expression B can be used to solve real-world problems. For example, in finance, we may need to calculate the interest rate on a loan or investment. In this case, we can use Expression A or Expression B to simplify the calculation.

Q: What are some common applications of Expression A and Expression B?

A: Some common applications of Expression A and Expression B include:

• Calculating interest rates on loans or investments
• Determining the value of a stock or bond
• Calculating the cost of goods sold or the value of inventory
• Determining the value of a property or asset

Q: How can I use Expression A and Expression B in my own work or studies?

A: To use Expression A and Expression B in your own work or studies, you can follow these steps:

1. Identify the problem or calculation you need to perform.
2. Determine whether Expression A or Expression B is the best choice for the problem.
3. Plug in the values and simplify the expression.
4. Use the simplified expression to solve the problem or calculate the answer.

Q: What are some common mistakes to avoid when using Expression A and Expression B?

A: Some common mistakes to avoid when using Expression A and Expression B include:

• Failing to check the conditions $m \neq 9$ and $m \neq -9$.
• Failing to simplify the expression correctly.
• Using the wrong expression for the problem.
• Failing to check the units or dimensions of the answer.

Q: How can I learn more about Expression A and Expression B?

A: To learn more about Expression A and Expression B, you can:

• Read online resources and tutorials.
• Watch video lectures or tutorials.
• Practice solving problems and exercises.
• Consult with a teacher or tutor.
• Read books or textbooks on the subject.

Q: What are some additional resources for learning about Expression A and Expression B?

A: Some additional resources for learning about Expression A and Expression B include:

• Khan Academy: Multiplying and dividing fractions
• Math Open Reference: Fractions
• Algebra.com: Simplifying fractions
• Purplemath: Fractions

Conclusion

In conclusion, Expression A and Expression B are two important expressions that can be used to simplify calculations and solve problems. By understanding the conditions and applications of these expressions, you can use them to solve real-world problems and improve your math skills.