MHB Solve Exponential Equation: 2+√2^x + 2-√2^x =4

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To solve the equation (2+√2)^x + (2-√2)^x = 4, the user introduces a substitution, letting C = (2+√2)^x. This leads to the conclusion that 1/C equals 1/(2-√2)^x, establishing a reciprocal relationship between the two terms. The discussion centers on understanding how this transformation simplifies the original equation. The user seeks clarification on the derivation of this reciprocal relationship. The conversation emphasizes the importance of algebraic manipulation in solving exponential equations.
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Hi, I'm having problems to solve this equation, pls help me:

$$\left( 2+\sqrt {2}\right) ^{x}+\left( 2-\sqrt {2}\right) ^{x}=4$$
 
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Chipset3600 said:
Hi, I'm having problems to solve this equation, pls help me:

$$\left( 2+\sqrt {2}\right) ^{x}+\left( 2-\sqrt {2}\right) ^{x}=4$$
Put $$C=(2+\sqrt {2})^x$$ Then $$\frac{1}{C}=\frac{1}{(2-\sqrt {2})^x}$$
NOW MY QUESTION IS, HOW DID I GET THIS?

Hint:
$$(2+\sqrt {2})^x (2-\sqrt {2})^x$$

Regards,
$$|\pi\rangle$$
 
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