\( 5(5)^{-x}-(5)^{-x-2}=24 {} 6 \quad 2^{\frac{x}{3}}+2^{\frac{x}{3}+1}=12 {} 7\left(5^{x}+10\right)\left(2^{x}-0,0625\right)=0 {} 8 \quad 2^{2 x}-3.2^{x}=-2 {} 9 \quad 9^{x}+6.3^{x}=27 {} 10 \quad 3^{2+x}+3^{2-x}=82 {} 115^{-x+1}+5^{1+x}=26 {} 122^{x+1}+2^{3-x}=17 {} 133^{x+2}-3^{x-2}=80 {} 142^{x}=\frac{27^{\frac{1}{2}}}{3^{\frac{21}{2}}+3^{\frac{1}{2}}} {} 156 x-5 x^{\frac{1}{2}}=6 {} 167^{x}-50=-49.7^{-x} {} 17 \quad 4.7^{4 x}=49.2^{4 x} {} 18 \quad 3^{3 x} \cdot 16=2^{3 x} \cdot 81 \)
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Beyond the Answer
Did you know that the concepts in these equations can actually trace their roots back to the ancient Greeks? Mathematicians like Euclid and Archimedes dabbled in the early forms of algebra, laying the groundwork for our modern understanding of equations and exponents. The exponential growth that comes into play here wasn’t just a mathematical curiosity; it was key in fields like astronomy, enabling ancient scholars to make sense of celestial movements! In the real world, we see exponential equations everywhere, from calculating compound interest in finance to population growth in ecology. For example, if you invest money at a certain interest rate, the money doesn't just grow linearly; it’s exponential! Understanding these equations can help you make informed decisions about saving, investing, and even predicting future trends in technology and social dynamics. So, embrace the power of exponents and see how they shape our world!