Which Expression Is Equivalent To $9(9m+3t)$?A. $18m+3t$B. \$81m+3i$[/tex\]C. $18m+12t$D. $81m+27t$

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that allows us to simplify complex expressions by multiplying each term inside the parentheses with the term outside. In this article, we will explore the distributive property and apply it to simplify the given expression $9(9m+3t)$.

The Distributive Property Formula

The distributive property formula is given by:

$$a(b+c) = ab + ac$$

where $a$, $b$, and $c$ are algebraic expressions.

Applying the Distributive Property

To simplify the expression $9(9m+3t)$, we will apply the distributive property by multiplying each term inside the parentheses with the term outside.

$$9(9m+3t) = 9 \times 9m + 9 \times 3t$$

Simplifying the Expression

Now, we will simplify the expression by multiplying the numbers and combining like terms.

$$9 \times 9m = 81m$$

$$9 \times 3t = 27t$$

Therefore, the simplified expression is:

$$81m + 27t$$

Comparing with the Options

Now, let's compare the simplified expression with the given options:

A. $18m+3t$ B. $81m+3t$ C. $18m+12t$ D. $81m+27t$

Conclusion

Based on the simplification, we can conclude that the expression equivalent to $9(9m+3t)$ is:

$$81m + 27t$$

Therefore, the correct answer is:

D. $81m+27t$

Real-World Applications

The distributive property is a powerful tool in algebra that has numerous real-world applications. It is used in various fields such as physics, engineering, and economics to simplify complex expressions and solve problems.

Practice Problems

Here are some practice problems to help you reinforce your understanding of the distributive property:

1. Simplify the expression $4(2x+5y)$.
2. Simplify the expression $3(7a-2b)$.
3. Simplify the expression $2(9c+4d)$.

Tips and Tricks

Here are some tips and tricks to help you master the distributive property:

1. Always multiply each term inside the parentheses with the term outside.
2. Simplify the expression by combining like terms.
3. Use the distributive property to simplify complex expressions.

Conclusion

In conclusion, the distributive property is a fundamental concept in algebra that allows us to simplify complex expressions by multiplying each term inside the parentheses with the term outside. By applying the distributive property, we can simplify the expression $9(9m+3t)$ to $81m + 27t$. We hope this article has provided you with a clear understanding of the distributive property and its applications in algebra.

Frequently Asked Questions

In this article, we will address some of the most frequently asked questions about the distributive property in algebra.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to simplify complex expressions by multiplying each term inside the parentheses with the term outside.

Q: How do I apply the distributive property?

A: To apply the distributive property, you need to multiply each term inside the parentheses with the term outside. For example, if you have the expression $a(b+c)$, you would multiply $a$ with $b$ and $c$ separately.

Q: What is the formula for the distributive property?

A: The formula for the distributive property is given by:

$$a(b+c) = ab + ac$$

where $a$, $b$, and $c$ are algebraic expressions.

Q: Can I use the distributive property with fractions?

A: Yes, you can use the distributive property with fractions. For example, if you have the expression $\frac{1}{2}(x+y)$, you would multiply $\frac{1}{2}$ with $x$ and $y$ separately.

Q: Can I use the distributive property with decimals?

A: Yes, you can use the distributive property with decimals. For example, if you have the expression $0.5(x+y)$, you would multiply $0.5$ with $x$ and $y$ separately.

Q: How do I simplify expressions using the distributive property?

A: To simplify expressions using the distributive property, you need to multiply each term inside the parentheses with the term outside and then combine like terms.

Q: Can I use the distributive property with negative numbers?

A: Yes, you can use the distributive property with negative numbers. For example, if you have the expression $-3(x+y)$, you would multiply $-3$ with $x$ and $y$ separately.

Q: Can I use the distributive property with variables?

A: Yes, you can use the distributive property with variables. For example, if you have the expression $2x(x+y)$, you would multiply $2x$ with $x$ and $y$ separately.

Q: What are some common mistakes to avoid when using the distributive property?

A: Some common mistakes to avoid when using the distributive property include:

• Not multiplying each term inside the parentheses with the term outside
• Not combining like terms
• Not using the correct formula for the distributive property

Q: How do I practice using the distributive property?

A: To practice using the distributive property, you can try simplifying expressions using the distributive property. You can also try solving problems that involve the distributive property.

Q: What are some real-world applications of the distributive property?

A: The distributive property has numerous real-world applications, including:

• Simplifying complex expressions in physics and engineering
• Solving problems in economics and finance
• Simplifying expressions in computer science and programming

Conclusion

In conclusion, the distributive property is a fundamental concept in algebra that allows us to simplify complex expressions by multiplying each term inside the parentheses with the term outside. By understanding the distributive property and its applications, you can become a more confident and proficient algebra student.