Final answer:
An irrational number can indeed be raised to an irrational power. This process involves understanding the rules of exponents and the manipulations of fractional exponents that represent roots, such as square or cube roots.
Step-by-step explanation:
Yes, you can raise an irrational number to an irrational power. While integer powers represent repeated multiplication, irrational exponents are a bit more complex. For instance, when raising a number to a power like 3¹.⁷, we approach this by taking the tenth-root of 3 (which is 3¹) and then raising that result to the 17th power, effectively calculating 3¹.⁷ as (3¹)¹⁷ or simply 3¹.
Understanding this concept relies on the rules of exponents outlined in mathematics. When we work with expressions such as x², which is equivalent to √x (the square root of x), we learn that fractional exponents represent roots.
Similarly, when we multiply or divide by numbers raised to a power, or when we raise a number with an exponent to another power, we use these rules to simplify or manipulate the expression.
Consider the Pythagorean Theorem; to solve for a side length denoted by 'a', we might need to take the square root to 'undo' the square, giving us 'a' raised to the power of 1 from 'a²'. Likewise, operations with units raised to a power involve manipulation of exponents, such as converting units or working with values in scientific notation without necessarily entering them into a calculator.