# Nakafa Framework: LLM

URL: https://nakafa.com/en/subject/high-school/10/mathematics/exponential-logarithm/radical-form
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/10/mathematics/exponential-logarithm/radical-form/en.mdx

Output docs content for large language models.

---

export const metadata = {
  title: "Radical Form",
  description: "Convert fractional exponents to radical form and vice versa. Master simplification techniques and understand key properties with real-world applications.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/04/2025",
  subject: "Exponential and Logarithm",
};

## Relationship Between Exponents and Radicals

Radical forms and exponents are closely related. When we have a number with a fractional exponent, we can convert it to a radical form.

Consider the following exponential function:

<BlockMath math="f(x) = 50(0.5)^x" />

This function represents the decay of drug dosage in a patient's body, where <InlineMath math="x" /> is the time needed for the drug to decay by half of its previous dosage.

If we want to know the amount of dosage that decays after 30 minutes, we substitute <InlineMath math="x = 30" /> into the function.

<BlockMath math="f(30) = 50(0.5)^{30}" />

For a time of half an hour, we can write the fractional exponent form:

<BlockMath math="f\left(\frac{1}{2}\right) = 50(0.5)^{\frac{1}{2}}" />

The fractional exponent <InlineMath math="(0.5)^{\frac{1}{2}}" /> is difficult to calculate manually. Therefore, we need an equivalent form.

Another form of <InlineMath math="(0.5)^{\frac{1}{2}}" /> is <InlineMath math="\sqrt{0.5}" />. This is what we call a radical form.

## Definition of Radical Form

The radical form is defined for any rational exponent <InlineMath math="\frac{m}{n}" />, where <InlineMath math="m" /> and <InlineMath math="n" /> are integers and <InlineMath math="n > 0" />:

<BlockMath math="a^{\frac{m}{n}} = (\sqrt[n]{a})^m \text{ or } a^{\frac{m}{n}} = \sqrt[n]{a^m}" />

This allows us to convert numbers with fractional exponents to radical form and vice versa.

## Simplifying Radical Forms

Here's an example of simplifying the multiplication of two radical forms:

Simplify the expression <InlineMath math="(2\sqrt{x})(3\sqrt[3]{x})" /> for <InlineMath math="x > 0" />

<BlockMath
  math="\begin{align}
(2\sqrt{x})(3\sqrt[3]{x}) &= (2x^{\frac{1}{2}})(3x^{\frac{1}{3}}) \\
&= 2 \cdot 3 \cdot x^{\frac{1}{2} + \frac{1}{3}} \\
&= 6x^{\frac{3+2}{6}} \\
&= 6x^{\frac{5}{6}} \\
&= 6\sqrt[6]{x^5}
\end{align}"
/>

Therefore, the simplified form is <InlineMath math="6x^{\frac{5}{6}}" /> or <InlineMath math="6\sqrt[6]{x^5}" />.

## Important Property of Radical Forms

It's important to understand that the form <InlineMath math="\sqrt{a+b} = \sqrt{a} + \sqrt{b}" /> is not correct.

Let's take the values <InlineMath math="a = 4" /> and <InlineMath math="b = 9" />, then:

<BlockMath math="\sqrt{4 + 9} = \sqrt{13} \neq \sqrt{4} + \sqrt{9} = 2 + 3 = 5" />

Another example:

<div className="flex flex-col gap-4">
  <BlockMath math="\sqrt{16 + 9} = \sqrt{25} = 5" />
  <BlockMath math="\sqrt{16} + \sqrt{9} = 4 + 3 = 7" />
</div>

Since <InlineMath math="5 \neq 7" />, it's clear that <InlineMath math="\sqrt{a+b} \neq \sqrt{a} + \sqrt{b}" />.