solve this equation: 9^x+1 + 1 = 10(3^x)

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Understand and Identify Bases: Understand the equation and identify similar bases. The equation is 9^x+1 + 1 = 10(3^x). We notice that 9 is a power of 3, specifically 9 = 3^2. This will allow us to rewrite the equation with a common base.Rewrite and Simplify Equation: Rewrite 9 as 3^2 and simplify the equation. 9^x+1 can be written as (3^2)^(x+1). Using the power of a power rule, we get 3^(2(x+1)). The equation now looks like this: 3^(2(x+1)) + 1 = 10(3^x)Expand Exponent: Expand the exponent in the term 3^(2(x+1)). 2(x+1) is equal to 2x + 2. So, we rewrite the equation as: 3^(2x+2) + 1 = 10(3^x)Divide and Simplify: Divide both sides of the equation by 3^x to simplify. (3^(2x+2))/(3^x) + 1/(3^x) = 10 Using the quotient of powers rule, we get: 3^(2x+2-x) + 1/(3^x) = 10 This simplifies to: 3^(x+2) + 1/(3^x) = 10Recognize Limitations: Recognize that the equation cannot be simplified further algebraically. At this point, we realize that the equation 3^(x+2) + 1/(3^x) = 10 does not lend itself to further algebraic manipulation. We need to use another method, such as trial and error, graphing, or a numerical method to find the value of x.Trial and Error Solution: Use trial and error or a numerical method to find the value of x. Since the problem does not specify a method, we will use trial and error. We know that 3^0 = 1, so let's try x = 0: 3^(0+2) + 1/(3^0) = 3^2 + 1 = 9 + 1 = 10 10 = 10 This is true, so x = 0 is a solution.

solve this equation: $9^x+1 + 1 = 10(3^x)$

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solve this equation: $9^x+1 + 1 = 10(3^x)$