# Nakafa Framework: LLM

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

Output docs content for large language models.

---

export const metadata = {
  title: "Property Proofs",
  description: "Learn rigorous proofs of 7 exponential properties with step-by-step demonstrations. Master multiplication, division, and power rules through examples.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/01/2025",
  subject: "Exponents and Logarithms",
};

## Exponential Properties and Their Proofs

We will discuss the properties of exponents and their proofs. These properties are essential to understand as they form the foundation for solving various problems related to exponents and logarithms.

### Property 1

<InlineMath math="a^m \cdot a^n = a^{m+n}" />

For every <InlineMath math="a \neq 0" /> and <InlineMath math="m, n" /> real numbers.

### Property 2

<InlineMath math="\frac{a^m}{a^n} = a^{m-n}" />

For every <InlineMath math="a \neq 0" /> and <InlineMath math="m, n" /> real numbers.

### Property 3

<InlineMath math="(a^m)^n = a^{m \cdot n}" />

For every <InlineMath math="a \neq 0" /> and <InlineMath math="m, n" /> real numbers.

### Property 4

<InlineMath math="(ab)^m = a^m \cdot b^m" />

For every <InlineMath math="a, b \neq 0" /> and <InlineMath math="m" /> integer.

**Proof:**

_Method 1:_

<div className="flex flex-col gap-4">
  <BlockMath math="(ab)^m = \underbrace{ab \times ab \times ab \times \ldots \times ab}_{m\text{ factors}}" />
  <BlockMath math="(ab)^m = \underbrace{a \times a \times a \times \ldots \times a}_{m\text{ factors}} \times \underbrace{b \times b \times b \times \ldots \times b}_{m\text{ factors}}" />
  <BlockMath math="(ab)^m = a^m \times b^m" />
</div>

_Method 2:_

We can take several examples with specific values of <InlineMath math="m, a" /> and <InlineMath math="b" />
to observe the pattern that forms. For example, with <InlineMath math="m=2, a=3, b=4" />:

<div className="flex flex-col gap-4">
  <BlockMath math="(3 \cdot 4)^2 = 12^2 = 144" />
  <BlockMath math="3^2 \cdot 4^2 = 9 \cdot 16 = 144" />
</div>

### Property 5

<InlineMath math="\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}" />

For every <InlineMath math="a \neq 0, b \neq 0" /> and <InlineMath math="m" /> integer.

**Proof:**

<div className="flex flex-col gap-4">
  <BlockMath math="\left(\frac{a}{b}\right)^m = \underbrace{\frac{a}{b} \times \frac{a}{b} \times \frac{a}{b} \times \ldots \times \frac{a}{b}}_{m\text{ factors}}" />
  <BlockMath math="\left(\frac{a}{b}\right)^m = \frac{\underbrace{a \times a \times a \times \ldots \times a}_{m\text{ factors}}}{\underbrace{b \times b \times b \times \ldots \times b}_{m\text{ factors}}}" />
  <BlockMath math="\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}" />
</div>

### Property 6

<InlineMath math="(a^{\frac{m}{n}})(a^{\frac{p}{n}}) = a^{\frac{m+p}{n}}" />

For every <InlineMath math="a > 0" />, <InlineMath math="\frac{m}{n}" /> and <InlineMath math="\frac{p}{n}" /> rational numbers with <InlineMath math="n \neq 0" />.

**Proof:**

Using Property 1, we obtain:

<div className="flex flex-col gap-4">
  <BlockMath math="(a^{\frac{m}{n}})(a^{\frac{p}{n}}) = a^{\frac{m}{n} + \frac{p}{n}}" />
  <BlockMath math="(a^{\frac{m}{n}})(a^{\frac{p}{n}}) = a^{\frac{m+p}{n}}" />
</div>

### Property 7

<InlineMath math="(a^{\frac{m}{n}})(a^{\frac{p}{q}}) = a^{\frac{mq + pn}{nq}}" />

For every <InlineMath math="a > 0" />, <InlineMath math="\frac{m}{n}" /> and <InlineMath math="\frac{p}{q}" /> rational numbers with <InlineMath math="n, q \neq 0" />.

**Proof:**

Using Property 1, we need to equalize the denominators of the exponents:

<div className="flex flex-col gap-4">
  <BlockMath math="(a^{\frac{m}{n}})(a^{\frac{p}{q}}) = a^{\frac{m}{n} + \frac{p}{q}}" />
  <BlockMath math="(a^{\frac{m}{n}})(a^{\frac{p}{q}}) = a^{\frac{mq}{nq} + \frac{pn}{nq}}" />
  <BlockMath math="(a^{\frac{m}{n}})(a^{\frac{p}{q}}) = a^{\frac{mq + pn}{nq}}" />
</div>

## Example Problems

### Determining Values and Proofs

1. <InlineMath math="(3^4)^2 = 3^p" />

    Solution: Using Property 3, <InlineMath math="(a^m)^n = a^{m \cdot n}" />

    <div className="flex flex-col gap-4">
      <BlockMath math="(3^4)^2 = 3^{4 \cdot 2} = 3^8" />
      <BlockMath math="p = 8" />
    </div>

2. <InlineMath math="b^4 \cdot b^5 = b^9" />

    Solution: Using Property 1, <InlineMath math="a^m \cdot a^n = a^{m+n}" />

    <div className="flex flex-col gap-4">
      <BlockMath math="b^4 \cdot b^5 = b^{4+5} = b^9" />
      <BlockMath math="\text{Therefore, the equation is proven true.}" />
    </div>

3. <InlineMath math="(3\pi)^p = 27\pi^3" />

    Solution: Converting <InlineMath math="27\pi^3" /> to <InlineMath math="3^3 \cdot \pi^3" />

    <div className="flex flex-col gap-4">
      <BlockMath math="27\pi^3 = 3^3 \cdot \pi^3 = (3\pi)^3" />
      <BlockMath math="p = 3" />
    </div>

### Simplifying Expressions

1. <InlineMath math="\left(\frac{2^4 \times 3^6}{2^3 \times 3^2}\right)^3" />

    <div className="flex flex-col gap-4 mt-4">
      <BlockMath math="\left(\frac{2^4 \times 3^6}{2^3 \times 3^2}\right)^3 = \left(2^{4-3} \times 3^{6-2}\right)^3" />
      <BlockMath math="= \left(2^1 \times 3^4\right)^3" />
      <BlockMath math="= 2^{1 \cdot 3} \times 3^{4 \cdot 3}" />
      <BlockMath math="= 2^3 \times 3^{12}" />
    </div>

2. <InlineMath math="(3u^3v^5)(9u^4v)" />

    <div className="flex flex-col gap-4 mt-4">
      <BlockMath math="(3u^3v^5)(9u^4v) = 3 \cdot 9 \cdot u^{3+4} \cdot v^{5+1}" />
      <BlockMath math="= 3^1 \cdot 3^2 \cdot u^7 \cdot v^6" />
      <BlockMath math="= 3^{1+2} \cdot u^7 \cdot v^6" />
      <BlockMath math="= 3^3 \cdot u^7 \cdot v^6" />
      <BlockMath math="= 27u^7v^6" />
    </div>

3. <InlineMath math="\left(\frac{n^{-1}r^4}{5n^{-6}r^{-4}}\right)^2" />

    <div className="flex flex-col gap-4 mt-4">
      <BlockMath math="\left(\frac{n^{-1}r^4}{5n^{-6}r^{-4}}\right)^2 = \left(\frac{n^{-1}r^4}{5} \cdot \frac{1}{n^{-6}r^{-4}}\right)^2" />
      <BlockMath math="= \left(\frac{n^{-1}r^4}{5} \cdot n^6r^4\right)^2" />
      <BlockMath math="= \left(\frac{n^{-1+6}r^{4+4}}{5}\right)^2" />
      <BlockMath math="= \left(\frac{n^5r^8}{5}\right)^2" />
      <BlockMath math="= \frac{n^{10}r^{16}}{25}" />
    </div>