2^sin3x*2^sin5x=2^sin4x

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To simplify this equation, we can use the property of exponents that states when multiplying two numbers with the same base, we can add the exponents:

2^(sin 3x) * 2^(sin 5x) = 2^(sin 3x + sin 5x)

Next, we can simplify the expression sin 3x + sin 5x by using the trigonometric identity sin(a) + sin(b) = 2sin((a+b)/2)cos((a-b)/2):

sin 3x + sin 5x = 2sin((3x+5x)/2)cos((5x-3x)/2) = 2sin(4x)cos(x).

Therefore, the original equation simplifies to:

2^(sin 3x) * 2^(sin 5x) = 2^(sin 4x)

becomes

2^(sin 3x + sin 5x) = 2^(sin 4x)

which simplifies to

2^(2sin(4x)cos(x)) = 2^(sin 4x).

Since the bases are the same, we can set the exponents equal to each other:

2sin(4x)cos(x) = sin 4x.

However, this equation cannot be simplified further without additional information, so the final simplified equation is:

2sin(4x)cos(x) = sin 4x.