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

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Grade 8

Solve equations with variable exponents

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Q. solve this equation:

9^x+1 + 1 = 10(3^x)

Understand and Identify Bases: Understand the equation and identify similar bases. $\newline$ 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. $\newline$ $9^x+1$ 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: $\newline$ $3^{2(x+1)} + 1 = 10(3^x)$
Expand Exponent: Expand the exponent in the term $3^{2(x+1)}$ . $\newline$ $2(x+1)$ is equal to $2x + 2$ . So, we rewrite the equation as: $\newline$ $3^{2x+2} + 1 = 10(3^x)$
Divide and Simplify: Divide both sides of the equation by $3^x$ to simplify. $\newline$ $(3^{2x+2})/(3^x) + 1/(3^x) = 10$ $\newline$ Using the quotient of powers rule, we get: $\newline$ $3^{2x+2-x} + 1/(3^x) = 10$ $\newline$ This simplifies to: $\newline$ $3^{x+2} + 1/(3^x) = 10$
Recognize Limitations: Recognize that the equation cannot be simplified further algebraically. At this point, we realize that the equation $3^{(x+2)} + \frac{1}{3^x} = 10$ 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 $x = 0$ : $3^{(0+2)} + \frac{1}{3^0} = 3^2 + 1 = 9 + 1 = 10$ $10 = 10$ This is true, so $x = 0$ is a solution.