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The problem asks for the simplified form of the following expression:

\[
(5^4) \times (5^3)^2 \div (5^5)^3
\]

Let's solve this step by step:

### Step 1: Simplify the powers
- Using the rule \((a^m)^n = a^{m \times n}\), we can simplify:
  \[
  (5^3)^2 = 5^{3 \times 2} = 5^6
  \]
  and
  \[
  (5^5)^3 = 5^{5 \times 3} = 5^{15}.
  \]

So now the expression becomes:
\[
5^4 \times 5^6 \div 5^{15}
\]

### Step 2: Apply the multiplication rule
- Using the rule \(a^m \times a^n = a^{m+n}\), we combine the first two terms:
  \[
  5^4 \times 5^6 = 5^{4+6} = 5^{10}.
  \]

Now the expression becomes:
\[
5^{10} \div 5^{15}.
\]

### Step 3: Apply the division rule
- Using the rule \(\frac{a^m}{a^n} = a^{m-n}\), we get:
  \[
  5^{10} \div 5^{15} = 5^{10-15} = 5^{-5}.
  \]

### Final Answer:
\[
5^{-5}.
\]

In decimal form, \(5^{-5} = \frac{1}{5^5} = \frac{1}{3125}\), but the simplified exponent form is \(5^{-5}\).

Would you like more details or have any questions about this solution?

Here are five related questions for further practice:
1. Simplify the expression \((2^3) \times (2^4) \div (2^5)\).
2. What is the result of \((7^2)^3 \div 7^4\)?
3. How do you simplify \((10^6) \times (10^3) \div (10^8)\)?
4. Find the simplified form of \((x^5) \times (x^2) \div (x^7)\).
5. How do you simplify an expression with negative exponents like \(2^{-3} \times 2^5\)?

**Tip:** Always apply exponent rules systematically—multiplication first, then division—when simplifying expressions with powers.

Learn how to simplify the expression (5^4) × (5^3)^2 ÷ (5^5)^3 step-by-step, using exponent rules. This solution explains power multiplication, division, and negative exponents, suitable for Grades 9-11.

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