Final answer:
A mathematical example of the expression a^(-m/n) = 1/a^(m/n) involves using a positive real number a and a rational exponent m/n.
Step-by-step explanation:
To explain the expression a^(-m/n) = 1/a^(m/n), let's consider an example where a is a positive real number and m/n is a rational number. Assume a = 16 and m/n = 2/4, which simplifies to 1/2.
Now, let's calculate a^(-m/n):
16^(-1/2) = 1/16^(1/2).
Here, the negative exponent indicates that the base (16) should be placed in the denominator of a fraction, and the exponent of 1/2 means we are looking for the square root.
So this simplifies to:
1/16^(1/2) = 1/4.
The reason 1/16^(1/2) simplifies to 1/4 is that the square root of 16 is 4, and the negative exponent has moved the base 16 to the denominator.
Therefore, a^(-m/n) equals 1/a^(m/n) because negative exponents flip the construction to the denominator, effectively showing that dividing by a number is equivalent to multiplying by its reciprocal.