https://file.aiquickdraw.com/mathbot/file/17256006368055pkxmdeu.jpg

The image contains a mathematical statement in Finnish, asking to prove a rule regarding powers of fractions. The statement is as follows:

**(A) Todista osamäärän potenssia koskeva laskusääntö**
\[
\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}, \quad \text{kun} \, b \neq 0.
\]

This is the power of a quotient rule in exponents, which states that raising a fraction to a power is the same as raising both the numerator and the denominator to that power, provided that the denominator is not zero.

### Proof:

To prove this, let's start with the definition:

\[
\left(\frac{a}{b}\right)^n = \frac{a}{b} \cdot \frac{a}{b} \cdot \frac{a}{b} \cdot \ldots \cdot \frac{a}{b} \quad \text{(n times)}
\]

By the properties of fractions, multiplying several fractions means multiplying all numerators together and all denominators together. So,

\[
\frac{a}{b} \cdot \frac{a}{b} \cdot \ldots \cdot \frac{a}{b} = \frac{a \cdot a \cdot \ldots \cdot a}{b \cdot b \cdot \ldots \cdot b}
\]

Thus,

\[
\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}
\]

This holds as long as \( b \neq 0 \) since division by zero is undefined.

Would you like more details or have any questions?

Here are five related questions to explore further:
1. What are the conditions under which the power of a quotient rule fails?
2. Can this rule be extended to negative exponents?
3. How does the power of a quotient rule relate to the laws of logarithms?
4. What happens when \( n = 0 \) in the context of the power of a quotient rule?
5. Can we prove this rule using logarithmic properties?

**Tip:** Understanding exponent rules is crucial for simplifying complex algebraic expressions efficiently!

(A) Todista osamäärän potenssia koskeva laskusääntö 
\( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \), kun \( b \neq 0 \).

This webpage provides a detailed proof of the power of a quotient rule for exponents, explaining how \( \left( \frac{a}{b} \right)^n \) is equivalent to \( \frac{a^n}{b^n} \), under the condition that \( b \neq 0 \). Suitable for students in Grades 9-12.

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