The equation in the image is:

$\frac{(b^{-2} \cdot (b^6)^{-2})}{b^{-5}}$

Let me simplify this step-by-step:

Step 1: Simplify the numerator

We have $b^{-2}$ and $(b^6)^{-2}$.

• For $(b^6)^{-2}$, apply the power of a power rule: $(a^m)^n = a^{m \cdot n}$. So, $(b^6)^{-2} = b^{6 \cdot (-2)} = b^{-12}$

• Now the expression in the numerator is $b^{-2} \cdot b^{-12}$.

• Applying the product of powers rule: $a^m \cdot a^n = a^{m + n}$, we get: $b^{-2} \cdot b^{-12} = b^{-2 + (-12)} = b^{-14}$

Step 2: Combine the expression

Now we have: $\frac{b^{-14}}{b^{-5}}$

Step 3: Apply the quotient rule

For division of powers with the same base: $\frac{a^m}{a^n} = a^{m - n}$. So, $\frac{b^{-14}}{b^{-5}} = b^{-14 - (-5)} = b^{-14 + 5} = b^{-9}$

Final Answer:

$b^{-9}$

Would you like a detailed explanation of any specific step or further clarifications?

Here are 5 related questions for further practice:

1. How would you simplify $\frac{a^7 \cdot a^{-3}}{a^2}$?
2. What is the result of $(x^4)^3 \cdot x^{-5}$?
3. Simplify the expression $\frac{y^6}{y^{-2}}$.
4. If $a = 2$, calculate the value of $a^{-4} \cdot a^2$.
5. What happens if you divide $b^{-5}$ by $b^{-2}$?

Tip: When working with powers, always remember to apply the rules for multiplication, division, and power of a power carefully!