Simplify a. \( \frac{2^{x+1}}{2^{x}} \)

When you simplify \( \frac{2^{x+1}}{2^{x}} \), you can use the property of exponents that states \( \frac{a^{m}}{a^{n}} = a^{m-n} \). Here, you subtract the exponents: \[ \frac{2^{x+1}}{2^{x}} = 2^{(x+1) - x} = 2^{1} = 2. \] So, the simplified expression is \( 2 \). Now, let’s dive into exponents! Did you know that the ancient Egyptians used a form of exponentiation in their mathematical texts? They utilized basic exponent rules to simplify calculations involving large numbers, particularly in the context of area and volume calculations. This fundamental understanding laid the groundwork for more complex mathematical concepts we use today. In real-life terms, think of exponents as a way to represent large amounts quickly! When dealing with things like population growth or radioactive decay, using exponentiation can make calculations manageable. Instead of writing out all the zeros, you can simply use powers to convey the same information more efficiently, saving time and reducing the risk of errors.