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GRE Number Theory
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Author:  questioner [ Sat Sep 22, 2012 5:21 am ]
Post subject:  GRE Number Theory

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I do not understand an example.

The example says "Factor: 3² + 12²" and provides the answer of 3²(1 + 4²).
This gave me 3² + 12^4.

If you look at the first problem in the example "Factor: 7² + 7^5", the provided answer of 7²(1 + 7³), this says that (7²)(7³) = 7^5. So why wouldn't (3²)(4²) = 12^4?


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Author:  Gennadiy [ Sat Sep 22, 2012 5:32 am ]
Post subject:  Re: GRE Number Theory

Quote:
If you look at the first problem in the example "Factor: 7² + 7^5", the provided answer of 7²(1 + 7³), this says that (7²)(7³) = 7^5. So why wouldn't (3²)(4²) = 12^4?
Take a look at these examples with factors instead of powers:

7² + 7^5 = 7 × 7 + 7 × 7 × 7 × 7 × 7 = (7 × 7)(1 + 7 × 7 × 7) = 7² (1 + 7³)

3² + 12² = 3 × 3 + (3 × 4)² = 3 × 3 + 3 × 3 × 4 × 4 = (3 × 3)(1 + 4 × 4) = 3² (1 + 4²)


Quote:
…, this says that (7²)(7³) = 7^5. So why wouldn't (3²)(4²) = 12^4?
In (7²)(7³) = 7^5 the factors have the same base, so we add the powers.
(7 × 7)(7 × 7 × 7) = 7^5

In (3²)(4²) = 12² the factors have the same powers, so we multiply the bases.
(3 × 3)(4 × 4) = (3 × 4)(3 × 4) = (3 × 4)² = 12²

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