Set Theory and Logic: Background You’ll Need  2

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Set Theory and Logic: Background You’ll Need 2

Simplify expressions using addition and multiplication

Commutative Properties

The commutative properties are fundamental rules of arithmetic that apply to the operations of addition and multiplication. These properties assert that the order in which two numbers are added or multiplied does not change the result. Such properties are essential for understanding the flexibility we have when rearranging and simplifying expressions in mathematics.

commutative properties

Commutative Property of Addition: if [latex]a[/latex] and [latex]b[/latex] are real numbers, then

[latex]a+b=b+a[/latex]

Commutative Property of Multiplication: if [latex]a[/latex] and [latex]b[/latex] are real numbers, then

[latex]a\cdot b=b\cdot a[/latex]

Associative Properties

Associative properties refer to the way numbers can be grouped within parentheses when added or multiplied without affecting the overall sum or product. These properties highlight that regardless of how we pair numbers, the result remains the same, simplifying computation and providing a basis for more advanced algebraic concepts.

associative properties

Associative Property of Addition: if [latex]a,b[/latex], and [latex]c[/latex] are real numbers, then

[latex]\left(a+b\right)+c=a+\left(b+c\right)[/latex]

Associative Property of Multiplication: if [latex]a,b[/latex], and [latex]c[/latex] are real numbers, then

[latex]\left(a\cdot b\right)\cdot c=a\cdot \left(b\cdot c\right)[/latex]

Simplify Expressions Using the Commutative and Associative Properties

Understanding the commutative and associative properties can greatly simplify the process of solving mathematical expressions. The commutative property allows us to rearrange numbers in addition or multiplication without changing the result, offering flexibility in how we approach problems. Meanwhile, the associative property lets us regroup numbers—also in addition or multiplication—ensuring that no matter how we pair the numbers, the outcome is consistent. Mastering these foundational rules will streamline your problem-solving, making complex calculations more manageable.

Simplify: [latex]-84n+\left(-73n\right)+84n[/latex]

Simplify: [latex]\Large\frac{7}{15}\cdot \frac{8}{23}\cdot \frac{15}{7}[/latex]

Simplify: [latex]\left(6.47q+9.99q\right)+1.01q[/latex]

Simplify: [latex]6\left(9x\right)[/latex]

	[latex]-84n+\left(-73n\right)+84n[/latex]
Re-order the terms.	[latex]-84n+84n+\left(-73n\right)[/latex]
Add left to right.	[latex]0+\left(-73n\right)[/latex]
Add.	[latex]-73n[/latex]

	[latex]\Large\frac{7}{15}\cdot \frac{8}{23}\cdot \frac{15}{7}[/latex]
Re-order the terms.	[latex]\Large\frac{7}{15}\cdot \frac{15}{7}\cdot \frac{8}{23}[/latex]
Multiply left to right.	[latex]1\cdot\Large\frac{8}{23}[/latex]
Multiply.	[latex]\Large\frac{8}{23}[/latex]

	[latex]\left(6.47q+9.99q\right)+1.01q[/latex]
Change the grouping.	[latex]6.47q+\left(9.99q+1.01q\right)[/latex]
Add in the parentheses first.	[latex]6.47q+\left(11.00q\right)[/latex]
Add.	[latex]17.47q[/latex]

	[latex]6\left(9x\right)[/latex]
Use the associative property of multiplication to re-group.	[latex]\left(6\cdot 9\right)x[/latex]
Multiply in the parentheses.	[latex]54x[/latex]