Answer:
1250.
Explanation:
To find the value of n in the equation 2^n-7 × 5^n-4 = 1250, we can simplify the equation step by step.
First, let's rewrite the equation in a more manageable form:
2^(n-7) × 5^(n-4) = 1250
Now, let's simplify the equation further:
(2^1)^(n-7) × (5^1)^(n-4) = 1250
2^(n-7) × 5^(n-4) = 1250
Next, let's simplify the left side of the equation by using the exponent properties:
2^(n-7) × 5^(n-4) = 2^3 × 5^3
2^(n-7) × 5^(n-4) = 8 × 125
2^(n-7) × 5^(n-4) = 1000
Since the bases of both terms on the left side are the same (2 and 5), we can equate the exponents:
n - 7 = 3
n - 4 = 3
Solving these equations separately, we find:
n = 10
n = 7
So, there are two possible values for n: 10 and 7.
However, it's important to note that we should check if these values satisfy the original equation. Plugging in n = 10:
2^(10-7) × 5^(10-4) = 1250
2^3 × 5^6 = 1250
8 × 15625 ≠ 1250
Therefore, n = 10 is not a valid solution.
Now, let's check n = 7:
2^(7-7) × 5^(7-4) = 1250
2^0 × 5^3 = 1250
1 × 125 = 1250
125 = 1250
This is false, so n = 7 is also not a valid solution.
Therefore, there is no value of n that satisfies the equation 2^n-7 × 5^n-4 = 1250.