Which Shows Two Expressions That Are Equivalent To ( − 7 ) ( − 15 ) ( − 5 (-7)(-15)(-5 ( − 7 ) ( − 15 ) ( − 5 ]?A. ( − 7 ) ( − 75 (-7)(-75 ( − 7 ) ( − 75 ] And ( − 1 ) ( 525 (-1)(525 ( − 1 ) ( 525 ] B. ( − 1 ) ( 525 (-1)(525 ( − 1 ) ( 525 ] And ( 35 ) ( − 15 (35)(-15 ( 35 ) ( − 15 ] C. ( 35 ) ( − 15 (35)(-15 ( 35 ) ( − 15 ] And ( 115 ) ( − 5 (115)(-5 ( 115 ) ( − 5 ] D. ( − 1 ) ( 525 (-1)(525 ( − 1 ) ( 525 ]

Introduction

When dealing with algebraic expressions, it's essential to understand the concept of equivalent expressions. Equivalent expressions are those that have the same value, even if they appear different. In this article, we will explore two expressions that are equivalent to $(-7)(-15)(-5)$ and examine the options provided to identify the correct pair of equivalent expressions.

What are Equivalent Expressions?

Equivalent expressions are algebraic expressions that have the same value, even if they appear different. They can be obtained by applying various mathematical operations, such as multiplication, division, addition, and subtraction. For example, the expressions $2x + 3$ and $3x + 2$ are equivalent because they have the same value for any given value of $x$.

Properties of Multiplication

To understand the concept of equivalent expressions, it's essential to recall the properties of multiplication. When multiplying two or more numbers, the order in which we multiply them does not change the result. This is known as the commutative property of multiplication. For example, $2 \times 3 = 3 \times 2 = 6$.

Associative Property of Multiplication

Another important property of multiplication is the associative property. This property states that when multiplying three or more numbers, we can regroup the numbers without changing the result. For example, $(2 \times 3) \times 4 = 2 \times (3 \times 4) = 24$.

Distributive Property of Multiplication

The distributive property of multiplication states that when multiplying a single number by two or more numbers, we can multiply each number separately and then add the results. For example, $2 \times (3 + 4) = 2 \times 3 + 2 \times 4 = 14$.

Applying the Properties of Multiplication

Now that we have a good understanding of the properties of multiplication, let's apply them to the given expression $(-7)(-15)(-5)$. We can use the associative property to regroup the numbers and simplify the expression.

Option A: $(-7)(-75)$ and $(-1)(525)$

Let's examine the first option, which states that $(-7)(-75)$ and $(-1)(525)$ are equivalent to $(-7)(-15)(-5)$. To verify this, we can multiply the numbers in each expression and compare the results.

$(-7)(-75) = 525$

$(-1)(525) = -525$

As we can see, the two expressions do not have the same value, so option A is incorrect.

Option B: $(-1)(525)$ and $(35)(-15)$

Let's examine the second option, which states that $(-1)(525)$ and $(35)(-15)$ are equivalent to $(-7)(-15)(-5)$. To verify this, we can multiply the numbers in each expression and compare the results.

$(-1)(525) = -525$

$(35)(-15) = -525$

As we can see, the two expressions have the same value, so option B is correct.

Option C: $(35)(-15)$ and $(115)(-5)$

Let's examine the third option, which states that $(35)(-15)$ and $(115)(-5)$ are equivalent to $(-7)(-15)(-5)$. To verify this, we can multiply the numbers in each expression and compare the results.

$(35)(-15) = -525$

$(115)(-5) = -575$

As we can see, the two expressions do not have the same value, so option C is incorrect.

Conclusion

In conclusion, we have examined the properties of multiplication and applied them to the given expression $(-7)(-15)(-5)$. We have also evaluated the options provided and identified the correct pair of equivalent expressions, which is option B: $(-1)(525)$ and $(35)(-15)$.

Final Answer

The final answer is option B: $(-1)(525)$ and $(35)(-15)$.

Introduction

In our previous article, we explored the concept of equivalent expressions and applied the properties of multiplication to simplify the expression $(-7)(-15)(-5)$. In this article, we will address some frequently asked questions (FAQs) on equivalent expressions to provide further clarification and understanding.

Q: What is the difference between equivalent expressions and identical expressions?

A: Equivalent expressions are those that have the same value, but may appear different. Identical expressions, on the other hand, are those that have the same appearance and value. For example, the expressions $2x + 3$ and $3x + 2$ are equivalent, but not identical.

Q: Can equivalent expressions be obtained by rearranging the terms?

A: Yes, equivalent expressions can be obtained by rearranging the terms. For example, the expressions $2x + 3$ and $3 + 2x$ are equivalent because they have the same value.

Q: Can equivalent expressions be obtained by multiplying or dividing both sides of an equation by a non-zero constant?

A: Yes, equivalent expressions can be obtained by multiplying or dividing both sides of an equation by a non-zero constant. For example, if we have the equation $2x = 4$, we can multiply both sides by $3$ to get $6x = 12$.

Q: Can equivalent expressions be obtained by adding or subtracting the same value to both sides of an equation?

A: Yes, equivalent expressions can be obtained by adding or subtracting the same value to both sides of an equation. For example, if we have the equation $2x = 4$, we can add $3$ to both sides to get $2x + 3 = 7$.

Q: How can we determine if two expressions are equivalent?

A: To determine if two expressions are equivalent, we can use the following steps:

1. Simplify both expressions by applying the properties of addition, subtraction, multiplication, and division.
2. Compare the simplified expressions to see if they have the same value.
3. If the simplified expressions have the same value, then the original expressions are equivalent.

Q: Can equivalent expressions be used to solve equations?

A: Yes, equivalent expressions can be used to solve equations. By simplifying both sides of an equation using equivalent expressions, we can isolate the variable and solve for its value.

Q: Are equivalent expressions always true?

A: No, equivalent expressions are not always true. If an expression is equivalent to a false statement, then it is also false. For example, the expression $2x = 4$ is equivalent to the expression $x = 2$, but the expression $2x = 5$ is not equivalent to the expression $x = 2.5$.

Conclusion

In conclusion, we have addressed some frequently asked questions (FAQs) on equivalent expressions to provide further clarification and understanding. Equivalent expressions are an essential concept in algebra, and understanding how to work with them can help us solve equations and simplify complex expressions.

Final Answer

The final answer is that equivalent expressions are those that have the same value, but may appear different, and can be obtained by applying the properties of addition, subtraction, multiplication, and division.