How do you simplify rational exponents?

Understand the Problem

The question is asking how to simplify expressions that involve rational exponents. This typically involves applying properties of exponents to rewrite the expression in a simpler form, which may include converting between radical and exponential forms.

Answer

The simplified expression for $x^{\frac{3}{2}}$ is $x \sqrt{x}$.

Steps to Solve

Identify the Expression with Rational Exponents

First, take a close look at the expression you want to simplify. Identify any parts of the expression that contain rational exponents. For example, if you have $x^{\frac{3}{2}}$, the rational exponent is $\frac{3}{2}$, which means you can rewrite it in radical form.

Convert Rational Exponents to Radical Form

The next step is to convert the rational exponents into radical form. The expression $x^{\frac{3}{2}}$ can be rewritten as:

$$ x^{\frac{3}{2}} = \sqrt{x^3} $$

This means you're taking the square root of $x^3$.

Simplify the Radical If Possible

Now, simplify the radical if possible. For example, $\sqrt{x^3}$ can be simplified further.

Using the property of square roots:

$$ \sqrt{x^3} = \sqrt{x^2 \cdot x} = \sqrt{x^2} \cdot \sqrt{x} = x \sqrt{x} $$

So, by simplifying, you find that:

$$ x^{\frac{3}{2}} = x \sqrt{x} $$

Combine the Steps

Put everything together to find the final simplified expression.

Now you have the expression in a simpler form.

The simplified expression for $x^{\frac{3}{2}}$ is $x \sqrt{x}$.

More Information

Rational exponents are a crucial part of algebra that can often make expressions more manageable. They provide a link between polynomial expressions and roots, allowing for easier manipulation and simplification.

Tips

Misinterpreting the rational exponent by forgetting to apply both the numerator and denominator correctly when converting to radical form.
Not simplifying the radical completely, leading to an incomplete answer.

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