Simplify the following: $\left[-(2 x-4 y)^6(2 x+4 y)^4\right]^3(2 x-4 y)^3(2 x+4 y)^6$
A. $(2 x-4 y)^{21}(2 x+4 y)^{18}$
B. $(2 x-4 y)^{12}(2 x+4 y)^{13}$
C. $-(2 x-4 y)^{21}(2 x+4 y)^{18}$
D. $-(2 x-4 y)^{12}(2 x+4 y)^{13}$
Answer (1)
Apply the power of a product rule to simplify the expression inside the brackets.
Multiply the result by the remaining terms.
Use the product of powers rule to combine exponents of like terms.
The simplified expression is − ( 2 x − 4 y ) 21 ( 2 x + 4 y ) 18 .
Explanation
Initial Analysis We are asked to simplify the expression [ − ( 2 x − 4 y ) 6 ( 2 x + 4 y ) 4 ] 3 ( 2 x − 4 y ) 3 ( 2 x + 4 y ) 6 . Let's break it down step by step.
Applying Power of a Product Rule First, we apply the power of a product rule to the expression inside the square brackets: ( ab ) n = a n b n . Also, remember that ( − 1 ) 3 = − 1 . So we have: [ − ( 2 x − 4 y ) 6 ( 2 x + 4 y ) 4 ] 3 = ( − 1 ) 3 ( 2 x − 4 y ) 6 × 3 ( 2 x + 4 y ) 4 × 3 = − ( 2 x − 4 y ) 18 ( 2 x + 4 y ) 12
Multiplying by Remaining Terms Now, we multiply this result by the remaining terms ( 2 x − 4 y ) 3 ( 2 x + 4 y ) 6 :
[ − ( 2 x − 4 y ) 6 ( 2 x + 4 y ) 4 ] 3 ( 2 x − 4 y ) 3 ( 2 x + 4 y ) 6 = − ( 2 x − 4 y ) 18 ( 2 x + 4 y ) 12 ( 2 x − 4 y ) 3 ( 2 x + 4 y ) 6
Combining Exponents Next, we use the product of powers rule: a m a n = a m + n . We combine the exponents of the ( 2 x − 4 y ) terms and the ( 2 x + 4 y ) terms: − ( 2 x − 4 y ) 18 ( 2 x + 4 y ) 12 ( 2 x − 4 y ) 3 ( 2 x + 4 y ) 6 = − ( 2 x − 4 y ) 18 + 3 ( 2 x + 4 y ) 12 + 6 = − ( 2 x − 4 y ) 21 ( 2 x + 4 y ) 18
Final Result So, the simplified expression is − ( 2 x − 4 y ) 21 ( 2 x + 4 y ) 18 .
Examples
Simplifying expressions like this is useful in many areas of math and science, such as when you're working with polynomials or complex equations in physics. For example, if you're modeling the motion of an object and you have an equation that involves terms like ( 2 x − 4 y ) and ( 2 x + 4 y ) , simplifying the equation can make it easier to understand and solve. This type of simplification helps in reducing the complexity of the equations, making them more manageable for further analysis or computation. It's like tidying up before starting a big project, making everything easier to handle.
