Use the definitions and theorems of this section to evaluate and simplify the following expression. Be sure to express answers with positive exponents.
(m . n)-1 =
I can only assume you mean
(m•n)^-1 = 1/(m•n)
To evaluate and simplify the expression (m . n)^-1, we need to use the definition of a negative exponent and apply the corresponding theorem.
The definition of a negative exponent states that any non-zero number raised to a negative exponent is equal to the reciprocal of the number raised to the positive exponent. In other words, x^-n = 1 / x^n.
Therefore, applying this definition to our expression, we have:
(m . n)^-1 = 1 / (m . n)^1
Now, we can simplify further using the theorem that states that the reciprocal of a product is equal to the product of the reciprocals. In other words, 1 / (a . b) = (1 / a) . (1 / b).
Applying this theorem to our expression, we have:
1 / (m . n)^1 = 1 / m^1 . 1 / n^1
Simplifying the exponents, we get:
1 / m^1 . 1 / n^1 = 1 / m . 1 / n
Finally, to express the answer with positive exponents, we can rewrite the reciprocal of a number with an exponent as a positive exponent. Therefore, 1 / m . 1 / n becomes m^-1 . n^-1.
So, the expression (m . n)^-1 simplifies to m^-1 . n^-1, with positive exponents.