https://file.aiquickdraw.com/mathbot/file/1727016269729vs3dx5nl.jpg

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!

Simplify the expression: \( \frac{(b^{-2} \cdot (b^6)^{-2})}{b^{-5}} \).

Learn how to simplify an expression involving powers and exponents: \( \frac{(b^{-2} \cdot (b^6)^{-2})}{b^{-5}} \). This problem explores key algebraic concepts like the laws of exponents, power of a power rule, and the quotient rule. Suitable for students in Grades 8-10.

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