Question: 2 and 3 with Exponent 7

This is a general property of all numbers.

Let's examine some similar examples:
8 + 8 + 8 = (3)(8)
1.1 + 1.1 + 1.1 = (3)(1.1)
20 + 20 + 20 = (3)(20)
x + x + x = 3x
k² + k² + k² = 3k²

Likewise, 3^7 + 3^7 + 3^7 = (3)(3^7)

Does that help?

You are suggesting that 3^7 + 3^7 + 3^7 = 9^7
So, you are adding the bases and keeping the exponents the same.
Unfortunately, this approach is not valid.

Let's examine some counter-examples that demonstrate that we can't just add the bases and keep the exponents the same:

1^5 + 1^5 + 1^5
Using your approach, we get: 1^5 + 1^5 + 1^5 = 3^5
Is this true?
No.
1^5 = 1
So, 1^5 + 1^5 + 1^5 = 1 + 1 + 1 = 3
However, your approach suggests that 1^5 + 1^5 + 1^5 = 3^5, yet 3^5 = 243

Or what about:
5^2 + 5^2
Using your approach, we get: 5^2 + 5^2 = 10^2
Is this true?
No.
5^2 = 25
So, 5^2 + 5^2 = 25 + 25 = 50
However, your approach suggests that 5^2 + 5^2 = 10^2 = 100

Does that help clear things up?

Cheers,
Brent

Unfortunately, there's no better approach.
This questions tests a very important general property of all numbers, so let's go over the solution.

First, here are some similar examples:
8 + 8 + 8 = (3)(8)
1.1 + 1.1 + 1.1 = (3)(1.1)
20 + 20 + 20 = (3)(20)
x + x + x = 3x
5w + 5w + 5w = (3)(5w) = 15w
k² + k² + k² = 3k²

Likewise, 3^7 + 3^7 + 3^7 = (3)(3^7)
The same concept applies to 2^7

Does that help?

Cheers,
Brent

Yours is a perfectly valid approach.

In my approach, I collected like terms to get: 3^7 + 3^7 + 3^7 = (3)(3^7), in the same way that k + k + k = 3k.

In your approach, you took 3^7 + 3^7 + 3^7 and factored out a 3^7 to get: 3^7(1 + 1 + 1)
When we simplify 3^7(1 + 1 + 1), we get (3)(3^7)

So, both approaches allow us to simplify 3^7 + 3^7 + 3^7 to get (3)(3^7)

Cheers,
Brent