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.