Question

2.1

Simplify fully, without the use of a calculator:
2.1.2
5^2𝑥+1+2.5^2𝑥−1 /2.5^2𝑥
(3)

Answer 1

[AI-generated answer — Bot GPT 3.5]

To simplify the given expression:

5^2𝑥+1 + 2.5^2𝑥-1 / 2.5^2𝑥

First, we can rewrite 2.5 as 5/2:

5^2𝑥+1 + (5/2)^2𝑥-1 / (5/2)^2𝑥

Now we can use the properties of exponents to simplify:

(5^(2𝑥) * 5^1 + 5^(2𝑥) * (5/2)^(-1)) / (5/2)^(2𝑥)

(5^(2𝑥+1) + 5^(2𝑥) * (2/5)^(1)) / (5/2)^(2𝑥)

(5^(2𝑥+1) + 5^(2𝑥) * 2/(5^2)) / (5/2)^(2𝑥)

(5^(2𝑥+1) + 5^(2𝑥) * 2/25) / (5/2)^(2𝑥)

Now we can simplify further if needed, but this is the fully simplified form of the expression.

Answer 2

how can you call that final mess "simplified"?

I suspect the question has been mangled, since
(5/2)^2𝑥-1 / (5/2)^2𝑥 = 5^-1 = 1/5
so the whole thing is just
5^2𝑥+1 + 1/5
I doubt this was the original intent

Answer 3

[AI-generated answer — Bot GPT 3.5]

Apologies for the confusion in my previous response. Let me simplify the expression correctly.

Given expression:
(5^2𝑥+1 + 2.5^2𝑥-1) / 2.5^2𝑥

Rewriting the expression with 2.5 as 5/2:

(5^(2𝑥+1) + (5/2)^(2𝑥-1)) / (5/2)^(2𝑥)

Using the property of exponents, we get:

(5^(2𝑥) * 5^1 + (5/2)^(2𝑥) / (5/2)^(2𝑥)

(5^(2𝑥) * 5 + (5/2)^(2𝑥) / (5/2)^(2𝑥)

(25 * 5^2𝑥 + (5/2)^(2𝑥) / (5/2)^(2𝑥)

(25 * 5^2𝑥 + 2^(2𝑥) * 5^(-2𝑥))

(25 * 5^(2𝑥) + 1) / 5

25 * 5^(2𝑥) / 5 + 1 / 5

5^(2𝑥) * 5 + 1 / 5

25 * 5 + 1 / 5

125 + 1 / 5

126 / 5

So, the simplified expression is 126/5. Thank you for catching the error and helping with the correction.