It looks like you are asking for expressions that can be proved equivalent by using certain properties of operations. Typically, the properties involved in proving the equivalence of expressions include:
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Associative Property: This property states that the way in which numbers are grouped does not change their sum or product.
- For addition: \( (a + b) + c = a + (b + c) \)
- For multiplication: \( (a \times b) \times c = a \times (b \times c) \)
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Commutative Property: This property indicates that the order of numbers does not affect their sum or product.
- For addition: \( a + b = b + a \)
- For multiplication: \( a \times b = b \times a \)
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Distributive Property: This property relates addition and multiplication, stating that multiplying a number by a sum is the same as multiplying each addend separately and then adding the results.
- \( a \times (b + c) = a \times b + a \times c \)
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Identity Property: This property refers to numbers that do not change others when added or multiplied.
- Additive identity: \( a + 0 = a \)
- Multiplicative identity: \( a \times 1 = a \)
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Inverse Property: This property deals with how a number and its opposite (additive) or reciprocal (multiplicative) come to a neutral element (zero or one, respectively).
- Additive inverse: \( a + (-a) = 0 \)
- Multiplicative inverse: \( a \times \frac{1}{a} = 1 \) (for \( a \neq 0 \))
Example of Equivalent Expressions
Here are a few examples that can be shown to be equivalent using the properties mentioned:
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Using the Commutative and Associative Properties
- Expression 1: \( 2 + 3 + 4 \)
- Expression 2: \( 4 + 3 + 2 \)
By the commutative property, we can rearrange the terms.
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Using the Distributive Property
- Expression 1: \( 3 \times (4 + 2) \)
- Expression 2: \( 3 \times 4 + 3 \times 2 \)
Here, we use the distributive property to show equality.
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Using the Associative Property and Identity Property
- Expression 1: \( (5 + 0) + 3 \)
- Expression 2: \( 5 + (0 + 3) \)
The associative property of addition allows us to rearrange the grouping.
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Using the Distributive Property and Commutative Property
- Expression 1: \( x(2 + 3) \)
- Expression 2: \( 5x \)
Applying the distributive property gives \( x \times 2 + x \times 3 \), which simplifies the original expression.
By applying these properties, you can identify many expressions that are equivalent to each other. If you have specific expressions in mind, please share them for a more tailored comparison!