$$ 5.2 ^ { x } + 3.2 ^ { x + 2 } = 68 $$

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To solve the exponential equation: $$ 5.2^x + 3.2^{x + 2} = 68 $$ Let's break it down step by step. ### Step 1: Simplify the Equation First, rewrite $$3.2^{x + 2}$$ using the properties of exponents: $$ 3.2^{x + 2} = 3.2^x \times 3.2^2 = 3.2^x \times 10.24 $$ Now, the equation becomes: $$ 5.2^x + 10.24 \times 3.2^x = 68 $$ ### Step 2: Set Up for Numerical Solution This equation involves different exponential bases ($$5.2$$ and $$3.2$$), making it challenging to solve algebraically. Therefore, we'll use numerical methods (such as the **Newton-Raphson method**) to approximate the value of $$x$$. ### Step 3: Define the Function Let's define the function $$f(x)$$ as: $$ f(x) = 5.2^x + 10.24 \times 3.2^x - 68 $$ We need to find $$x$$ such that $$f(x) = 0$$. ### Step 4: Initial Estimates Let's test some initial values to narrow down the range where $$x$$ lies. - **For $$x = 1$$:** $$ f(1) = 5.2 + 10.24 \times 3.2 \approx 5.2 + 32.768 = 37.968 < 68 $$ - **For $$x = 2$$:** $$ f(2) = 5.2^2 + 10.24 \times 3.2^2 \approx 27.04 + 104.8576 = 131.8976 > 68 $$ Thus, $$x$$ lies between 1 and 2. ### Step 5: Refine the Estimate Continuing with more precise estimates: - **For $$x = 1.5$$:** $$ f(1.5) \approx 11.8576 + 58.534 \approx 70.392 > 68 $$ - **For $$x = 1.45$$:** $$ f(1.45) \approx 10.887 + 55.29 \approx 66.177 < 68 $$ Since $$f(1.45) < 68$$ and $$f(1.5) > 68$$, the solution lies between 1.45 and 1.5. Using linear approximation and iterative methods, we refine the estimate: $$ x \approx 1.4718 $$ ### Step 6: Final Solution Thus, the solution to the equation $$5.2^x + 3.2^{x + 2} = 68$$ is approximately: $$ x \approx 1.47 $$ ### Verification To ensure accuracy, plug $$x \approx 1.47$$ back into the original equation: $$ 5.2^{1.47} + 10.24 \times 3.2^{1.47} \approx 11.24 + 56.78 \approx 68.02 \approx 68 $$ This confirms that our approximation is correct. **Final Answer:** The solution is approximately $$ x \approx 1.47 $$. The solution is approximately $$ x \approx 1.47 $$.

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