%5E2%2B8*5%5Ex-2*25%5Ex%2B15.jpeg "(25^x-4*5^x)^2+8*5^x 2*25^x+15")
Simplifying the Expression: (25^x - 45^x)^2 + 85^x * 2*25^x + 15
This problem involves simplifying a complex expression with exponential terms. We can achieve this through a series of algebraic manipulations, taking advantage of the properties of exponents. Here's a step-by-step solution:
1. Factor out 5^x:
Notice that both 25^x and 5^x can be expressed in terms of 5^x:
- 25^x = (5^2)^x = 5^(2x)
Therefore, the expression can be rewritten as:
(5^(2x) - 45^x)^2 + 85^x * 2*5^(2x) + 15
2. Simplify the expression:
Now, let's simplify the expression further by factoring out a common factor of 5^x:
- (5^x(5^x - 4))^2 + 85^x * 25^(2x) + 15
- 5^(2x) (5^x - 4)^2 + 16*5^(3x) + 15
3. Expand the square term:
Expand the square term to get:
- 5^(2x) (5^(2x) - 85^x + 16) + 165^(3x) + 15
4. Distribute and combine like terms:
Distribute 5^(2x) and combine like terms:
- 5^(4x) - 85^(3x) + 165^(2x) + 16*5^(3x) + 15
- 5^(4x) + 85^(3x) + 165^(2x) + 15
Final Simplified Expression:
The simplified form of the expression is:
(5^(4x) + 85^(3x) + 165^(2x) + 15)
This expression can be further factored, but it depends on the context of the problem. In some cases, the simplified form above might be sufficient.