Simplify fifth root of m^5n^-5

Solution:

Step 1:

Apply the rule for negative exponents: $a^{-n} = \frac{1}{a^n}$. Thus, we have $\sqrt[5]{m^5 \cdot \frac{1}{n^5}}$.

Step 2:

Merge the terms $m^5$ and $\frac{1}{n^5}$ under a single radical: $\sqrt[5]{\frac{m^5}{n^5}}$.

Step 3:

Express $\frac{m^5}{n^5}$ as a single fraction raised to the fifth power: $\sqrt[5]{\left(\frac{m}{n}\right)^5}$.

Step 4:

Extract the fifth root of the fifth power, which simplifies to the base of the power, assuming all variables represent real numbers: $\frac{m}{n}$.

Knowledge Notes:

The problem involves simplifying a radical expression with both positive and negative exponents. The relevant knowledge points include:

1. Negative Exponent Rule: For any nonzero number $b$ and integer $n$, $b^{-n} = \frac{1}{b^n}$. This rule is used to transform negative exponents into positive exponents within a fraction.

2. Radical and Exponent Relationship: For any nonnegative real number $a$ and positive integer $n$, $\sqrt[n]{a^n} = a$ when $a$ is a real number. This property allows us to simplify expressions where the index of the radical and the exponent of the term under the radical are the same.

3. Fractional Exponents: The expression $\sqrt[n]{a}$ can be rewritten as $a^{\frac{1}{n}}$. In this problem, we use the inverse property, where $a^{\frac{n}{n}} = a$.

4. Simplification of Radicals: When the index of the radical and the exponent of the number inside the radical are the same, the radical sign and the exponent cancel each other out, leaving the base of the exponent.

Using these principles, we can simplify the given expression by first converting the negative exponent to a positive one, then combining the terms under the radical, and finally applying the relationship between radicals and exponents to simplify the expression to $\frac{m}{n}$.