Name The Property The Equation Illustrates.$2\left(-\frac{3}{9}\right)=\left(-\frac{3}{9}\right)^2$A. Commutative Property Of Addition B. Inverse Property Of Multiplication C. Commutative Property Of Multiplication D. Associative Property

The world of algebra is filled with various properties that help us simplify and solve equations. In this article, we will delve into one of these properties, which is illustrated by the given equation: $2\left(-\frac{3}{9}\right)=\left(-\frac{3}{9}\right)^2$. Our goal is to identify the property that this equation represents.

What are Algebraic Properties?

Algebraic properties are rules that govern the behavior of numbers and variables in mathematical operations. They help us simplify expressions, solve equations, and understand the relationships between different mathematical concepts. There are four main types of algebraic properties: commutative, associative, distributive, and inverse.

Commutative Property of Addition

The commutative property of addition states that the order of the numbers being added does not change the result. In other words, $a + b = b + a$. This property is true for all real numbers.

Commutative Property of Multiplication

The commutative property of multiplication states that the order of the numbers being multiplied does not change the result. In other words, $a \cdot b = b \cdot a$. This property is also true for all real numbers.

Associative Property

The associative property states that the order in which we perform operations does not change the result. In other words, $(a + b) + c = a + (b + c)$. This property is true for addition and multiplication.

Distributive Property

The distributive property states that a single operation can be distributed over multiple operations. In other words, $a(b + c) = ab + ac$.

Inverse Property of Addition

The inverse property of addition states that for every number $a$, there exists a number $-a$ such that $a + (-a) = 0$.

Inverse Property of Multiplication

The inverse property of multiplication states that for every number $a$, there exists a number $\frac{1}{a}$ such that $a \cdot \frac{1}{a} = 1$.

Analyzing the Given Equation

Now that we have a good understanding of the different algebraic properties, let's analyze the given equation: $2\left(-\frac{3}{9}\right)=\left(-\frac{3}{9}\right)^2$. To identify the property that this equation represents, we need to simplify the equation and see which property is being illustrated.

Simplifying the Equation

To simplify the equation, we can start by evaluating the expression inside the parentheses: $-\frac{3}{9}$. This can be simplified to $-\frac{1}{3}$.

Simplifying the Left Side of the Equation

Now that we have simplified the expression inside the parentheses, we can simplify the left side of the equation: $2\left(-\frac{1}{3}\right)$. This can be simplified to $-\frac{2}{3}$.

Simplifying the Right Side of the Equation

The right side of the equation is already simplified: $\left(-\frac{1}{3}\right)^2$. This can be simplified to $\frac{1}{9}$.

Comparing the Simplified Equations

Now that we have simplified both sides of the equation, we can compare them: $-\frac{2}{3} = \frac{1}{9}$. Unfortunately, these two expressions are not equal, which means that the given equation is not true.

Conclusion

In conclusion, the given equation $2\left(-\frac{3}{9}\right)=\left(-\frac{3}{9}\right)^2$ does not illustrate any of the algebraic properties that we discussed earlier. The equation is not true, and therefore, it does not represent any of the properties.

Answer

The correct answer is None of the above. The given equation does not illustrate any of the algebraic properties that we discussed earlier.

Final Thoughts

In our previous article, we discussed the different algebraic properties and analyzed a given equation to see which property it represents. However, we received many questions from readers who wanted to know more about these properties. In this article, we will answer some of the most frequently asked questions about algebraic properties.

Q: What is the commutative property of addition?

A: The commutative property of addition states that the order of the numbers being added does not change the result. In other words, $a + b = b + a$. This property is true for all real numbers.

Q: What is the commutative property of multiplication?

A: The commutative property of multiplication states that the order of the numbers being multiplied does not change the result. In other words, $a \cdot b = b \cdot a$. This property is also true for all real numbers.

Q: What is the associative property?

A: The associative property states that the order in which we perform operations does not change the result. In other words, $(a + b) + c = a + (b + c)$. This property is true for addition and multiplication.

Q: What is the distributive property?

A: The distributive property states that a single operation can be distributed over multiple operations. In other words, $a(b + c) = ab + ac$.

Q: What is the inverse property of addition?

A: The inverse property of addition states that for every number $a$, there exists a number $-a$ such that $a + (-a) = 0$.

Q: What is the inverse property of multiplication?

A: The inverse property of multiplication states that for every number $a$, there exists a number $\frac{1}{a}$ such that $a \cdot \frac{1}{a} = 1$.

Q: How do I apply these properties in real-life situations?

A: Algebraic properties are used in many real-life situations, such as finance, science, and engineering. For example, when calculating the total cost of items, you can use the commutative and associative properties to simplify the calculation. When solving equations, you can use the inverse property to isolate the variable.

Q: Can I use these properties to solve equations with variables?

A: Yes, you can use algebraic properties to solve equations with variables. For example, if you have an equation like $2x + 3 = 5$, you can use the inverse property to isolate the variable $x$.

Q: What are some common mistakes to avoid when using algebraic properties?

A: Some common mistakes to avoid when using algebraic properties include:

• Not following the order of operations (PEMDAS)
• Not simplifying expressions before applying properties
• Not checking the validity of the equation before applying properties

Q: How can I practice using algebraic properties?

A: You can practice using algebraic properties by working on problems that involve these properties. You can also use online resources, such as math websites and apps, to practice and review these concepts.

Conclusion

In conclusion, algebraic properties are essential tools for simplifying and solving equations. By understanding and applying these properties, you can solve problems more efficiently and effectively. We hope that this article has helped you to better understand these properties and how to use them in real-life situations.