Here's one approach.....

Key concept: \((x^n)(y^n) = (xy)^n\)

Given:
Quantity A: \(5^x +1\)
Quantity B: \(6^x\)

Subtract \(5^x\) from both quantities to get:
Quantity A: \(1\)
Quantity B: \(6^x - 5^x\)

Applied the key concept above to factor 5^x out of Quantity B to get:
Quantity A: \(1\)
Quantity B: \(5^x(1.2^x - 1)\)

At this point we can apply a little bit of number sense...
Since \(x>1\), we know that \(5^x\) is greater than 5
Since \(x>1\), we know that \(1.2^x\) is greater than 1.2, which means \(1.2^x - 1\) is greater than 0.2

So Quantity B becomes:
Quantity A: \(1\)
Quantity B: (some number greater than 5)(some number greater than 0.2)

We know that \((5)(0.2) = 1\), so when we simplify Quantity B we get:
Quantity A: \(1\)
Quantity B: some product greater than 1

Answer: B

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Is that explanation complete ??
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Yes, I was just seeing if I my coding was correct.
It's complete now.
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Given that \(x > 1\), we need to compare:

Quantity A: \(5^x + 1\)
Quantity B: \(6^x\)


\(5^x + 1\) VS \(6^x\)

Divide both sides by \(5^x\):

\((5^x + 1)/5^x\) VS \(6^x/5^x\)

Or \((1 + 1/5^x)\) VS \((6/5)^x\)

Or \((1 + 1/5^x)\) VS \((1 + 1/5)^x\)

The term \((1 + 1/5)^x\) can be expanded using binomial expansion to get: \(1^x + ... \) (many terms in-between) \( ... + (1/5)^x\)

Thus, Quantity B is greater than Quantity A

Answer B
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