6. If [tex]p =\frac{-4}{9}, q =\frac{2}{3}, r =\frac{-8}{11}[/tex]. Verify the following:
a) [tex]p+(q+r)=(p+q)+r[/tex]
b) [tex]p \times q=q \times p[/tex]
c) [tex]p \times(q+r)=(p \times q)+(p \times r)[/tex]
Answer (2)
Verify the associative property of addition: p + ( q + r ) = ( p + q ) + r , both sides equal 99 − 50 .
Verify the commutative property of multiplication: p × q = q × p , both sides equal 27 − 8 .
Verify the distributive property of multiplication over addition: p × ( q + r ) = ( p × q ) + ( p × r ) , both sides equal 297 8 .
All three properties are verified.
Explanation
Problem Setup We are given p = 9 − 4 , q = 3 2 , and r = 11 − 8 . We need to verify the associative and commutative properties, and the distributive property.
Verifying Associative Property of Addition a) We need to verify if p + ( q + r ) = ( p + q ) + r . Let's calculate both sides separately.
First, we calculate q + r : q + r = 3 2 + 11 − 8 = 33 22 − 33 24 = 33 − 2 Now, we calculate p + ( q + r ) : p + ( q + r ) = 9 − 4 + 33 − 2 = 99 − 44 − 99 6 = 99 − 50 Next, we calculate p + q : p + q = 9 − 4 + 3 2 = 9 − 4 + 9 6 = 9 2 Now, we calculate ( p + q ) + r : ( p + q ) + r = 9 2 + 11 − 8 = 99 22 − 99 72 = 99 − 50 Since both sides are equal, the associative property of addition is verified.
Verifying Commutative Property of Multiplication b) We need to verify if p × q = q × p . Let's calculate both sides separately.
First, we calculate p × q : p × q = 9 − 4 × 3 2 = 27 − 8 Now, we calculate q × p : q × p = 3 2 × 9 − 4 = 27 − 8 Since both sides are equal, the commutative property of multiplication is verified.
Verifying Distributive Property of Multiplication over Addition c) We need to verify if p × ( q + r ) = ( p × q ) + ( p × r ) . Let's calculate both sides separately.
First, we calculate q + r : q + r = 3 2 + 11 − 8 = 33 22 − 33 24 = 33 − 2 Now, we calculate p × ( q + r ) : p × ( q + r ) = 9 − 4 × 33 − 2 = 297 8 Next, we calculate p × q : p × q = 9 − 4 × 3 2 = 27 − 8 Now, we calculate p × r : p × r = 9 − 4 × 11 − 8 = 99 32 Now, we calculate ( p × q ) + ( p × r ) : ( p × q ) + ( p × r ) = 27 − 8 + 99 32 = 297 − 88 + 297 96 = 297 8 Since both sides are equal, the distributive property of multiplication over addition is verified.
Conclusion All three properties are verified with the given values of p , q , and r .
Examples
These properties are fundamental in algebra and arithmetic. For example, the associative property ensures that when calculating the total cost of multiple items with discounts, the order in which you add the costs and discounts doesn't change the final amount. The commutative property is used when calculating areas or volumes, where the order of multiplying dimensions doesn't affect the result. The distributive property is essential in calculating taxes or tips, where you apply a percentage to a sum of values.
All three properties involving the values of p, q, and r are verified: the associative property of addition holds, the commutative property of multiplication is confirmed, and the distributive property of multiplication over addition is validated. Each property calculation shows equal results for both sides of the equations. Thus, the operations performed are consistent with the mathematical properties being tested.
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