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Find the value of A that makes the following equation true for all values of x. {:[3^(x)-3^(x-2)=A*3^(x)],[A=]:}

Find the value of A A that makes the following equation true for all values of x x .\newline3x−3x−2=A⋅3xA= \begin{array}{l} 3^{x}-3^{x-2}=A \cdot 3^{x} \\ A= \end{array}

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Q. Find the value of A A that makes the following equation true for all values of x x .\newline3x−3x−2=A⋅3xA= \begin{array}{l} 3^{x}-3^{x-2}=A \cdot 3^{x} \\ A= \end{array}
  1. Given equation: We are given the equation:\newline3x−3x−2=A⋅3x3^{x} - 3^{x-2} = A\cdot3^{x}\newlineWe need to find the value of AA that makes this equation true for all xx.
  2. Simplifying the left side: First, let's simplify the left side of the equation by factoring out the common term 3x3^{x}:\newline3x−3x⋅3−2=A⋅3x3^{x} - 3^{x}\cdot3^{-2} = A\cdot3^{x}
  3. Rewriting 3−23^{-2}: Now, we can rewrite 3−23^{-2} as 1/321/3^2, which is 1/91/9: 3x−3x∗(1/9)=A∗3x3^{x} - 3^{x}*(1/9) = A*3^{x}
  4. Factoring out 3x3^{x}: Next, we can factor out 3x3^{x} from both terms on the left side:\newline3x×(1−19)=A⋅3x3^{x} \times (1 - \frac{1}{9}) = A\cdot3^{x}
  5. Simplifying the expression: Now, we simplify the expression in the parentheses:\newline1−19=891 - \frac{1}{9} = \frac{8}{9}\newlineSo, we have:\newline3x×(89)=A×3x3^{x} \times \left(\frac{8}{9}\right) = A\times3^{x}
  6. Equating coefficients: Since we want the equation to be true for all xx, we can equate the coefficients of 3x3^{x} on both sides of the equation:\newline89=A\frac{8}{9} = A
  7. Value of A: Therefore, the value of AA that makes the equation true for all xx is 89\frac{8}{9}.

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