What is a rational exponent and how does it relate to radical expressions?

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A rational exponent is an exponent that can be expressed as a fraction, typically in the form $$ \frac{m}{n} $$, where $$ m $$ and $$ n $$ are integers and $$ n $$ is not zero. The expression $$ a^{\frac{m}{n}} $$ can be interpreted in two ways:

1. **Root and Power**: The expression $$ a^{\frac{m}{n}} $$ can be rewritten as $$ \sqrt[n]{a^m} $$. This means that you first take the $$ n $$-th root of $$ a $$ and then raise the result to the power of $$ m $$.

2. **Power and Root**: Alternatively, it can also be expressed as $$ (a^m)^{\frac{1}{n}} $$, which means you first raise $$ a $$ to the power of $$ m $$ and then take the $$ n $$-th root of that result.

### Relationship to Radical Expressions

Radical expressions are another way to represent roots. For example, the square root of $$ a $$ can be written as $$ \sqrt{a} $$, which is equivalent to $$ a^{\frac{1}{2}} $$. More generally, the $$ n $$-th root of $$ a $$ is expressed as $$ \sqrt[n]{a} $$, which corresponds to $$ a^{\frac{1}{n}} $$.

Thus, there is a direct relationship between rational exponents and radical expressions:

- $$ a^{\frac{1}{n}} = \sqrt[n]{a} $$
- $$ a^{\frac{m}{n}} = \sqrt[n]{a^m} = (a^m)^{\frac{1}{n}} $$

### Examples

1. $$ a^{\frac{1}{3}} = \sqrt[3]{a} $$ (the cube root of $$ a $$)
2. $$ a^{\frac{2}{5}} = \sqrt[5]{a^2} $$ (the fifth root of $$ a^2 $$)

In summary, rational exponents provide a convenient way to express roots and powers in a unified manner, allowing for easier manipulation of expressions involving both roots and exponents.
A rational exponent is a fraction that represents both a root and a power. For example, $$ a^{\frac{1}{2}} $$ is the same as the square root of $$ a $$. This means that rational exponents allow you to express roots and powers in a single expression, making it easier to work with them in math problems.

What is a rational exponent and how does it relate to radical expressions?

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