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Find the value of A that makes the following equation true for all values of x. {:[5^(x)+5^(x+3)=A*5^(x)],[A=]:}

Find the value of A A that makes the following equation true for all values of x x .\newline5x+5x+3=A⋅5xA= \begin{array}{l} 5^{x}+5^{x+3}=A \cdot 5^{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 .\newline5x+5x+3=A⋅5xA= \begin{array}{l} 5^{x}+5^{x+3}=A \cdot 5^{x} \\ A= \end{array}
  1. Given Equation Transformation: We are given the equation 5x+5x+3=A⋅5x5^{x} + 5^{x+3} = A\cdot5^{x}. To find the value of AA, we need to express the left side of the equation in terms of 5x5^{x} so that we can compare it directly to A⋅5xA\cdot5^{x}.
  2. Exponent Property Application: We can rewrite 5(x+3)5^{(x+3)} as 5x⋅535^{x}\cdot5^{3} because of the property of exponents that states a(b+c)=ab⋅aca^{(b+c)} = a^{b} \cdot a^{c}.
  3. Substitution and Simplification: Substitute 5(x+3)5^{(x+3)} with 5x⋅535^{x}\cdot5^{3} in the original equation to get 5x+5x⋅53=A⋅5x5^{x} + 5^{x}\cdot5^{3} = A\cdot5^{x}.
  4. Factor Out Common Term: Now, factor out 5x5^{x} from the left side of the equation to get 5x(1+53)=A⋅5x5^{x}(1 + 5^{3}) = A\cdot5^{x}.
  5. Calculate 535^3: Calculate 5(3)5^{(3)} which is 5∗5∗5=1255*5*5 = 125.
  6. Final Substitution: Substitute 535^{3} with 125125 in the equation to get 5x(1+125)=A⋅5x5^{x}(1 + 125) = A\cdot5^{x}.
  7. Combine Terms: Add 1+1251 + 125 to get 126126. So the equation now is 5x⋅126=A⋅5x5^{x}\cdot126 = A\cdot5^{x}.
  8. Coefficient Comparison: Since the equation must be true for all values of xx, we can equate the coefficients of 5x5^{x} on both sides of the equation. This gives us 126=A126 = A.
  9. Final Value of A: We have found the value of AA to be 126126, which makes the original equation true for all values of xx.

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