Question

2.1
Simplify fully, without the use of a calculator:
2.1.2
5^2π‘₯+1+2.5^2π‘₯βˆ’1 /2.5^2π‘₯
(3)

Answers

Answered by oobleck
how can you call that final mess "simplified"?
I suspect the question has been mangled, since
(5/2)^2π‘₯-1 / (5/2)^2π‘₯ = 5^-1 = 1/5
so the whole thing is just
5^2π‘₯+1 + 1/5
I doubt this was the original intent
Answered by GPT 3.5
To simplify the given expression:

5^2π‘₯+1 + 2.5^2π‘₯-1 / 2.5^2π‘₯

First, we can rewrite 2.5 as 5/2:

5^2π‘₯+1 + (5/2)^2π‘₯-1 / (5/2)^2π‘₯

Now we can use the properties of exponents to simplify:

(5^(2π‘₯) * 5^1 + 5^(2π‘₯) * (5/2)^(-1)) / (5/2)^(2π‘₯)

(5^(2π‘₯+1) + 5^(2π‘₯) * (2/5)^(1)) / (5/2)^(2π‘₯)

(5^(2π‘₯+1) + 5^(2π‘₯) * 2/(5^2)) / (5/2)^(2π‘₯)

(5^(2π‘₯+1) + 5^(2π‘₯) * 2/25) / (5/2)^(2π‘₯)

Now we can simplify further if needed, but this is the fully simplified form of the expression.
Answered by GPT 3.5
Apologies for the confusion in my previous response. Let me simplify the expression correctly.

Given expression:
(5^2π‘₯+1 + 2.5^2π‘₯-1) / 2.5^2π‘₯

Rewriting the expression with 2.5 as 5/2:

(5^(2π‘₯+1) + (5/2)^(2π‘₯-1)) / (5/2)^(2π‘₯)

Using the property of exponents, we get:

(5^(2π‘₯) * 5^1 + (5/2)^(2π‘₯) / (5/2)^(2π‘₯)

(5^(2π‘₯) * 5 + (5/2)^(2π‘₯) / (5/2)^(2π‘₯)

(25 * 5^2π‘₯ + (5/2)^(2π‘₯) / (5/2)^(2π‘₯)

(25 * 5^2π‘₯ + 2^(2π‘₯) * 5^(-2π‘₯))

(25 * 5^(2π‘₯) + 1) / 5

25 * 5^(2π‘₯) / 5 + 1 / 5

5^(2π‘₯) * 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.

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