Use the properties of exponents to rewrite the expression. (3x⁴y³)(-5x³y)
A. 15x⁷y
B. -15x⁷y
C. 15x¹²y³
D. -15x⁷y⁴
Answer (2)
Multiply the coefficients: 3 × − 5 = − 15 .
Multiply the x terms: x 4 ⋅ x 3 = x 4 + 3 = x 7 .
Multiply the y terms: y 3 ⋅ y = y 3 + 1 = y 4 .
Combine the results: − 15 x 7 y 4 . The correct answer is − 15 x 7 y 4 .
Explanation
Understanding the Problem We are asked to rewrite the expression ( 3 x 4 y 3 ) ( − 5 x 3 y ) using the properties of exponents. The key properties we'll use are:
x a ⋅ x b = x a + b (when multiplying powers with the same base, add the exponents)
( c ⋅ x ) ( d ⋅ y ) = c ⋅ d ⋅ x ⋅ y (the order of multiplication doesn't matter)
Multiplying the Coefficients First, let's multiply the coefficients (the numerical parts) together:
3 × − 5 = − 15
Multiplying the x Terms Next, let's multiply the x terms together. We have x 4 and x 3 . Using the property x a ⋅ x b = x a + b , we get:
x 4 ⋅ x 3 = x 4 + 3 = x 7
Multiplying the y Terms Now, let's multiply the y terms together. We have y 3 and y . Remember that y is the same as y 1 . So, using the property y a ⋅ y b = y a + b , we get:
y 3 ⋅ y 1 = y 3 + 1 = y 4
Combining the Results Finally, let's combine all the parts together. We have the coefficient − 15 , the x term x 7 , and the y term y 4 . So the simplified expression is:
− 15 x 7 y 4
Final Answer Therefore, the rewritten expression is − 15 x 7 y 4 . Comparing this to the given options, we see that the correct answer is not among them. However, if the original expression was ( 3 x 4 y 3 ) ( − 5 x 3 y 1 ) , then the correct answer would be − 15 x 7 y 4 . There seems to be a typo in the options provided. Assuming the options are incorrect and we need to choose the closest one, option B is closest if the exponent of y was a typo. However, the correct answer based on our calculations is − 15 x 7 y 4 .
Examples
Understanding how to simplify expressions with exponents is crucial in many areas of mathematics and science. For instance, in physics, when calculating the force between two charged particles, you often encounter expressions involving exponents. Suppose the force F is given by F = k r 2 q 1 q 2 , where k is a constant, q 1 and q 2 are the charges, and r is the distance between them. If you double the charges and halve the distance, the new force F ′ becomes F ′ = k ( 2 r ) 2 ( 2 q 1 ) ( 2 q 2 ) = k 4 r 2 4 q 1 q 2 = 16 k r 2 q 1 q 2 = 16 F . This shows how understanding exponents helps in quickly determining how changes in variables affect the outcome.
The expression ( 3 x 4 y 3 ) ( − 5 x 3 y ) simplifies to − 15 x 7 y 4 using the properties of exponents. Therefore, the correct answer is option B. -15x⁷y⁴.
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