Verify the following • 15 x [7 + (-3)] = [15 x 7] + [15 x (-3)] • [(-21) x (-4)] + [(-21) x (-6)] = (-21) x [(-4) + (-6)]
The Correct Answer and Explanation is:
Let’s verify both of these algebraic expressions:
1. 15 x [7 + (-3)] = [15 x 7] + [15 x (-3)]
This is an example of the distributive property of multiplication over addition.
- Left-hand side:
15×(7+(−3))15 \times (7 + (-3))15×(7+(−3))
15×(7−3)=15×4=6015 \times (7 – 3) = 15 \times 4 = 6015×(7−3)=15×4=60 - Right-hand side:
(15×7)+(15×−3)(15 \times 7) + (15 \times -3)(15×7)+(15×−3)
105+(−45)=105−45=60105 + (-45) = 105 – 45 = 60105+(−45)=105−45=60
Since both sides are equal (60=6060 = 6060=60), the equation is true.
2. [(-21) x (-4)] + [(-21) x (-6)] = (-21) x [(-4) + (-6)]
This is another example of the distributive property applied to two terms.
- Left-hand side:
(−21)×(−4)+(−21)×(−6)(-21) \times (-4) + (-21) \times (-6)(−21)×(−4)+(−21)×(−6)
84+126=21084 + 126 = 21084+126=210 (since multiplying two negatives gives a positive result) - Right-hand side:
(−21)×(−4+−6)=(−21)×(−10)(-21) \times (-4 + -6) = (-21) \times (-10)(−21)×(−4+−6)=(−21)×(−10)
210210210 (multiplying two negatives results in a positive)
Since both sides are equal (210=210210 = 210210=210), this equation is also true.
Explanation:
- The distributive property states that for any numbers aaa, bbb, and ccc, the equation a×(b+c)=(a×b)+(a×c)a \times (b + c) = (a \times b) + (a \times c)a×(b+c)=(a×b)+(a×c) holds.
- In both of these expressions, the distributive property is used to simplify and break down the multiplication and addition operations, which leads to the verification of the equality.
