Write a video explaining if 1^1 = 1^2, then why 1 ≠ 2 graphically and mathematically

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Here's an interesting mathematical puzzle. We have the equation one to the first power equals one to the second power. Let's evaluate both sides. One to the first power equals one, and one to the second power also equals one. So we get one equals one, which is definitely true. But this raises a question: if the equation is true, why doesn't this mean that the exponents one and two are equal? The key to understanding this puzzle lies in a fundamental rule about exponents. Generally, if a to the x equals a to the y, then x must equal y. However, this rule has important exceptions when the base a is zero, one, or negative one. When the base is one, one raised to any power always equals one. For example, one to the hundredth power equals one, one to the negative fifth power equals one, and even one to the pi power equals one. This is why the function y equals one to the x is simply the horizontal line y equals one, unlike other exponential functions like y equals two to the x which actually grow. Now let's analyze this graphically. The equation one to the x equals one squared asks us to find where these two functions intersect. The function y equals one to the x simplifies to y equals one for all values of x, which is a horizontal line. The function y equals one squared is simply the constant one, also a horizontal line at y equals one. When we graph both functions, we see they are exactly the same line! This means they intersect at every point, not just at specific x values. The equation is true for all x, including x equals one and x equals two, but this doesn't mean one equals two. To better understand why this happens with base one, let's compare it with base two. If we have the equation two to the x equals two squared, this equation has only one solution: x equals two. This is because the function y equals two to the x is strictly increasing, so it intersects the horizontal line y equals four at exactly one point. In contrast, when the base is one, the function y equals one to the x is constant at y equals one. This constant function intersects any horizontal line at the same height everywhere, giving us infinite solutions. Therefore, one to the first power equals one squared doesn't imply that one equals two, because the base one is a special case where the exponential function becomes constant. To summarize what we've learned: The equation one to the first power equals one squared is indeed true because both sides equal one. However, this doesn't mean the exponents are equal. The general rule that if a to the x equals a to the y then x equals y fails when the base is zero, one, or negative one. When the base is one, the exponential function becomes a constant horizontal line, which intersects any other horizontal line at the same height everywhere, not just at one point. This is why mathematical rules often have important exceptions that we must carefully consider.