https://file.aiquickdraw.com/mathbot/file/1726187498081k34otyj7.jpg

The given expression is:

\[
\frac{a^5 \cdot b^{-2} \cdot c^3}{(bc)^3}
\]

with the values provided:
- \(a = 3\)
- \(b = 4\)
- \(c = 1\)

Let's simplify the expression first and then substitute the values.

### Step 1: Simplifying the expression
We know:
\[
(bc)^3 = b^3 \cdot c^3
\]

So the expression becomes:
\[
\frac{a^5 \cdot b^{-2} \cdot c^3}{b^3 \cdot c^3}
\]

Now, simplify the terms:
- \(b^{-2} \div b^3 = b^{-2-3} = b^{-5}\)
- \(c^3 \div c^3 = 1\) (the \(c^3\) terms cancel out)

Thus, the expression simplifies to:
\[
a^5 \cdot b^{-5}
\]

### Step 2: Substituting the values
Now, substitute \(a = 3\) and \(b = 4\):

\[
(3)^5 \cdot (4)^{-5} = 243 \cdot \frac{1}{1024}
\]

\[
\frac{243}{1024}
\]

This is the simplified result. 

Would you like further explanation on any of the steps?

### Related questions:
1. How would the expression change if \(b\) were a negative number?
2. Can you express \(\frac{243}{1024}\) as a decimal?
3. How does changing the exponent of \(b\) impact the final result?
4. What happens to the expression if \(c\) were not equal to 1?
5. How do negative exponents affect division in general?

**Tip:** When simplifying powers, always combine the exponents of like bases by adding or subtracting them as per the rules of exponents!

Simplify and evaluate the expression a^5 * b^(-2) * c^3 / (b * c)^3 with a = 3, b = 4, c = 1.

Learn how to simplify the algebraic expression a^5 * b^(-2) * c^3 / (b * c)^3 with step-by-step evaluation. The problem covers key exponent rules, suitable for Grades 8-10.

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