Solving a fraction with an exponent in the denominator involves simplifying the expression, often by manipulating the exponent or rationalizing the denominator. The specific approach depends on the form of the exponent (integer or fractional).
Understanding the Basics
Before diving into specific scenarios, let's clarify some foundational concepts:
- Exponent Rules: Remember key rules like ( x^{-n} = \frac{1}{x^n} ) and ( \frac{x^a}{x^b} = x^{a-b} ).
- Fractional Exponents: A fractional exponent like ( x^{\frac{1}{n}} ) represents the nth root of x. For example, ( x^{\frac{1}{2}} = \sqrt{x} ) and ( x^{\frac{1}{3}} = \sqrt[3]{x} ).
Scenarios and Solutions
Here are a few common scenarios and how to address them:
1. Integer Exponent in the Denominator
If you have a fraction like ( \frac{1}{x^n} ), where n is an integer, you can rewrite it using a negative exponent:
( \frac{1}{x^n} = x^{-n} )
Example:
( \frac{1}{x^2} = x^{-2} )
2. Fractional Exponent in the Denominator
This is where things get more interesting, and the YouTube video "Fractional Exponents in Denominator" becomes relevant. If you have something like ( \frac{1}{x^{\frac{1}{n}}} ), you can still use the negative exponent rule:
( \frac{1}{x^{\frac{1}{n}}} = x^{-\frac{1}{n}} )
However, it's often preferable to rationalize the denominator. Let's say you have ( \frac{1}{x^{\frac{1}{3}}} ). Here’s how you can approach it:
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Rewrite with a negative exponent: ( x^{-\frac{1}{3}} )
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Rationalize (optional, but often helpful): To eliminate the fractional exponent in the denominator (or, equivalently, make the negative exponent positive), multiply both the numerator and denominator by a suitable power of x. In this case, multiply by ( \frac{x^{\frac{2}{3}}}{x^{\frac{2}{3}}} ):
( \frac{1}{x^{\frac{1}{3}}} \cdot \frac{x^{\frac{2}{3}}}{x^{\frac{2}{3}}} = \frac{x^{\frac{2}{3}}}{x^{\frac{1}{3} + \frac{2}{3}}} = \frac{x^{\frac{2}{3}}}{x^1} = \frac{x^{\frac{2}{3}}}{x} )
Example Inspired by the Reference:
The reference discusses simplifying expressions like ( \frac{x^{\frac{1}{3}}}{x} ). This is already in a simplified form but can be further expressed as:
( \frac{x^{\frac{1}{3}}}{x} = x^{\frac{1}{3} - 1} = x^{-\frac{2}{3}} = \frac{1}{x^{\frac{2}{3}}} )
Alternatively, you can interpret ( x^{\frac{1}{3}} ) as the cube root of x:
( \frac{x^{\frac{1}{3}}}{x} = \frac{\sqrt[3]{x}}{x} )
Summary Table
| Original Expression | Step 1: Rewrite | Step 2: Simplify/Rationalize | Final Result |
|---|---|---|---|
| ( \frac{1}{x^2} ) | ( x^{-2} ) | N/A | ( x^{-2} ) |
| ( \frac{1}{x^{\frac{1}{3}}} ) | ( x^{-\frac{1}{3}} ) | Multiply by ( \frac{x^{\frac{2}{3}}}{x^{\frac{2}{3}}} ) | ( \frac{x^{\frac{2}{3}}}{x} ) |
| ( \frac{x^{\frac{1}{3}}}{x} ) | N/A | Subtract exponents | ( x^{-\frac{2}{3}} ) or ( \frac{\sqrt[3]{x}}{x} ) |
