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

Recognize Perfect Cube: The function given is g(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.Factor Sum of Cubes: The expression inside the square root is indeed a sum of cubes: (2^(3x)+1)^(3) + (2x)^(3). The sum of cubes can be factored using the identity a^3 + b^3 = (a + b)(a^2 - ab + b^2).Apply Sum of Cubes Identity: Let's apply the sum of cubes identity to our expression with a = 2^(3x)+1 and b = 2x. This gives us (2^(3x)+1 + 2x)((2^(3x)+1)^2 - (2^(3x)+1)(2x) + (2x)^2).Simplify Factored Expression: Now we simplify the expression inside the square root by factoring it. We get sqrt((2^(3x)+1 + 2x)((2^(3x)+1)^2 - (2^(3x)+1)(2x) + (2x)^2)).Simplify Square Root: Since we are taking the square root of a product of two terms, and one of the terms is the sum (2^(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) simplifies to 2^(3x)+1 + 2x.

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

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Algebra 2

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

Recognize Perfect Cube: The function given is $g(x) = \sqrt{(2^{3x}+1)^{3} + (2x)^{3}}$ . 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.
Factor Sum of Cubes: The expression inside the square root is indeed a sum of cubes: $(2^{3x}+1)^{3} + (2x)^{3}$ . The sum of cubes can be factored using the identity $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ .
Apply Sum of Cubes Identity: Let's apply the sum of cubes identity to our expression with $a = 2^{3x}+1$ and $b = 2x$ . This gives us $(2^{3x}+1 + 2x)((2^{3x}+1)^2 - (2^{3x}+1)(2x) + (2x)^2)$ .
Simplify Factored Expression: Now we simplify the expression inside the square root by factoring it. We get $\sqrt{(2^{3x}+1 + 2x)((2^{3x}+1)^2 - (2^{3x}+1)(2x) + (2x)^2)}$ .
Simplify Square Root: Since we are taking the square root of a product of two terms, and one of the terms is the sum $2^{3x}+1 + 2x$ , 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, $g(x)$ simplifies to $2^{3x}+1 + 2x$ .