Understanding the Exponential Equation 2^(2x+y) = (4^x)(2^y)

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The equation 2^(2x+y) = (4^x)(2^y) holds true due to the properties of exponents. By substituting values for x and y, the equivalence can be verified, demonstrating that both sides yield the same result. The discussion highlights the rules of combining exponents, specifically x^a * x^b = x^(a+b) and x^(ab) = (x^a)^b. Understanding these rules clarifies why the initial equation is valid. This foundational knowledge is essential for solving similar exponential equations.
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Is 22x+y = 4x2y? If I substitute various digits into the x and y variables, it works, but I can't understand why. Can anyone please explain this to me?

For example, if we choose x=3 and y=1,
2(2)(3)+1 = 27 = 128
2(2)(3)+1 = 4321 = 64(2) = 128

Thanks
 
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It follows from the basic rules of combining exponents.

x^a x^b = x^{(a+b)}

x^{ab} = (x^a)^b
 
I can't believe I didn't figure that out. Thank you!
 

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