The expression in the image is:

$\frac{3^{-5} \times 10^{-5} \times 125}{5^{-7} \times 6^{-5}}$

Let's simplify it step by step.

Step 1: Simplify powers of 125

Since $125 = 5^3$, we can rewrite $125$ as $5^3$: $\frac{3^{-5} \times 10^{-5} \times 5^3}{5^{-7} \times 6^{-5}}$

Step 2: Combine like terms

Now we handle the powers of $5$ and other terms separately:

• For $5$, we have $5^3$ in the numerator and $5^{-7}$ in the denominator. Using the property $\frac{a^m}{a^n} = a^{m-n}$, we get: $5^{3 - (-7)} = 5^{3 + 7} = 5^{10}$
• The other terms stay the same for now: $\frac{3^{-5} \times 10^{-5} \times 5^{10}}{6^{-5}}$

Step 3: Combine remaining terms

At this point, we have no other like bases, so the final simplified expression is:

$3^{-5} \times 10^{-5} \times \frac{5^{10}}{6^{-5}} = \frac{5^{10} \times 6^5}{3^5 \times 10^5}$

This is the fully simplified form of the given expression. Would you like further clarification or details on any step?

Here are 5 related questions:

1. How do you simplify expressions with negative exponents?
2. How do you apply the laws of exponents to expressions with multiple terms?
3. What is the general formula for dividing terms with the same base and different exponents?
4. How do you handle numbers expressed as powers of prime factors in such equations?
5. Can you simplify complex fractions further if no common factors are present?

Tip: When working with exponents, always combine terms with the same base first before simplifying the entire expression!