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g(x)=sqrt((2^(3x)+1)^(3)+(2x)^(3))

g(x)=(23x+1)3+(2x)3 g(x)=\sqrt{\left(2^{3 x}+1\right)^{3}+(2 x)^{3}}

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Full solution

Q. g(x)=(23x+1)3+(2x)3 g(x)=\sqrt{\left(2^{3 x}+1\right)^{3}+(2 x)^{3}}
  1. Recognize Perfect Cube: The function given is g(x)=(23x+1)3+(2x)3g(x) = \sqrt{(2^{3x}+1)^{3} + (2x)^{3}}. To simplify this, we need to recognize if the expression inside the square root is a perfect cube or can be factored into a sum of cubes.
  2. Factor Sum of Cubes: The expression inside the square root is indeed a sum of cubes: (23x+1)3+(2x)3(2^{3x}+1)^{3} + (2x)^{3}. The sum of cubes can be factored using the identity a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2).
  3. Apply Sum of Cubes Identity: Let's apply the sum of cubes identity to our expression with a=23x+1a = 2^{3x}+1 and b=2xb = 2x. This gives us (23x+1+2x)((23x+1)2−(23x+1)(2x)+(2x)2)(2^{3x}+1 + 2x)((2^{3x}+1)^2 - (2^{3x}+1)(2x) + (2x)^2).
  4. Simplify Factored Expression: Now we simplify the expression inside the square root by factoring it. We get (23x+1+2x)((23x+1)2−(23x+1)(2x)+(2x)2)\sqrt{(2^{3x}+1 + 2x)((2^{3x}+1)^2 - (2^{3x}+1)(2x) + (2x)^2)}.
  5. Simplify Square Root: Since we are taking the square root of a product of two terms, and one of the terms is the sum 23x+1+2x2^{3x}+1 + 2x, we can simplify the square root to just this sum, because it is the square root of the product of two identical factors (which is the definition of a square root). Therefore, g(x)g(x) simplifies to 23x+1+2x2^{3x}+1 + 2x.

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