Simplify the expression using the Product Property.
$\left(5 a^{-6} b^6\right)\left(4 a^9 b^{-5}\right)=$
Answer (1)
Multiply the coefficients: 5 × 4 = 20 .
Multiply the terms with 'a': a − 6 × a 9 = a 3 .
Multiply the terms with 'b': b 6 × b − 5 = b .
Combine the results to get the final simplified expression: 20 a 3 b .
Explanation
Understanding the problem We are given the expression ( 5 a − 6 b 6 ) ( 4 a 9 b − 5 ) and we need to simplify it using the product property of exponents. The product property states that x m × x n = x m + n .
Multiplying the coefficients First, we multiply the coefficients: 5 × 4 = 20 .
Multiplying the 'a' terms Next, we multiply the terms with 'a': a − 6 × a 9 = a − 6 + 9 = a 3 .
Multiplying the 'b' terms Then, we multiply the terms with 'b': b 6 × b − 5 = b 6 + ( − 5 ) = b 1 = b .
Combining the results Finally, we combine the results: 20 × a 3 × b = 20 a 3 b . Therefore, the simplified expression is 20 a 3 b .
Examples
Imagine you are calculating the volume of a rectangular prism where the dimensions involve exponents. Simplifying expressions like the one above helps in easily computing the volume without dealing with complex exponents. For instance, if the sides of the prism are 5 a − 6 b 6 , 4 a 9 b − 5 , and a 2 b , the volume would involve multiplying these terms, which simplifies to 20 a 5 b 2 . This is useful in various fields like engineering and architecture where dimensions and volumes need to be calculated efficiently.
