5⁶+2²×25³+2³×125²/26×5⁶

To solve the expression 5⁶ + 2² × 25³ + 2³ × 125² / 26 × 5⁶, we will follow the order of operations: parentheses, exponents, multiplication and division from left to right, and addition and subtraction from left to right.

First, let's convert the terms that involve powers of 25 and 125 into terms involving base 5 and base 2, since 25 = 5² and 125 = 5³.

1. Calculate the components involving exponents:
- 25³ = (5²)³ = 5^(2*3) = 5^6
- 125² = (5³)² = 5^(3*2) = 5^6

Thus, our expression can be rewritten:
- 2² × 25³ = 2² × 5^6
- 2³ × 125² = 2³ × 5^6

Now substitute these back into the original expression:

5^6 + 2² × 5^6 + 2³ × 5^6 / 26 × 5^6

2. Factor out 5^6 from the terms:
- The first term is 5^6.
- The second term is 2² × 5^6.
- The third term (before dividing) can be seen as (2³ × 5^6).

This simplifies to:
5^6 + (2² + 2³) × 5^6

Now, we can combine the 5^6 terms:
Let’s find the common factor:
= 5^6 [1 + 2² + 2³]

3. Calculate 2² and 2³:
- 2² = 4
- 2³ = 8

Now substitute these values back:
= 5^6 [1 + 4 + 8]
= 5^6 [13]

So now we have:
= 13 × 5^6

4. Evaluate division by 26:
Now we divide this whole expression by 26:
(13 × 5^6) / 26
= (13 / 26) × 5^6
= (1/2) × 5^6

5. Final answer in a simpler exponential form:
5^6 = (5²)³ = 25³ = 15625

Thus, we have:
(1/2) × 15625
= 7812.5

The answer to the exercise 5⁶ + 2² × 25³ + 2³ × 125² / 26 × 5⁶ is 7812.5.