Question

six to the power of negative twelve over nine to the power of negative fithteen equals nine to the power of fithteen over six to the power of twelve

Answer 1

which rule is it

Answer 2

To simplify the expression \( \frac{6^{-12}}{9^{-15}} \) and check if it equals \( \frac{9^{15}}{6^{12}} \), we can rewrite the negative exponents and manipulate the fractions.

Starting with the left-hand side:

\[
\frac{6^{-12}}{9^{-15}} = 6^{-12} \cdot 9^{15}
\]

The right-hand side is:

\[
\frac{9^{15}}{6^{12}} = 9^{15} \cdot 6^{-12}
\]

Both sides are the same:

\[
6^{-12} \cdot 9^{15} = 9^{15} \cdot 6^{-12}
\]

This clearly shows that both sides are identical since they are equal to \( 9^{15} \cdot 6^{-12} \).

Therefore, the original statement can be confirmed:

\[
\frac{6^{-12}}{9^{-15}} = \frac{9^{15}}{6^{12}}
\]

This holds true. Hence, the equation is valid.

Answer 3

The rules used to simplify and verify the equation involve properties of exponents, particularly:

1. **Negative Exponent Rule**:
- \( a^{-n} = \frac{1}{a^n} \)
- This implies that negative exponents indicate a reciprocal.

2. **Multiplication of Exponents**:
- \( a^m \cdot a^n = a^{m+n} \)
- This rule allows us to combine terms when they have the same base.

3. **Division of Exponents**:
- \(\frac{a^m}{a^n} = a^{m-n}\)
- Like the multiplication rule, this allows combining or simplifying terms in a fraction.

In your specific case:

- You used the **Negative Exponent Rule** to rewrite the terms.
- You then observed that each expression could be rearranged to show they are equivalent by recognizing that \( 6^{-12} \) and \( 9^{-15} \) could be rewritten.

So, the overall approach was primarily based on the **Negative Exponent Rule** and properties of multiplication and division of exponents.