Exponents Question!

[alexander.vien]

I'm having trouble understanding the following exponents simplification question:

Simplify: m^n-2 + 3m^(n) - 4m^(n+1)


Answer: [spoiler]m^n-2 is the GCF of all the terms - factor it out to obtain m^n-2(1 + 3m^2 - 4m^3)[/spoiler]


How is that the GCF? I don't get how you factor that out of all the other terms.

Thank you in advance!

Alex[spoiler][/spoiler]

[Brent@GMATPrepNow]

Let's begin with some non-variable exponents.
For example, to factor x^8 + x^4 - x^3, we recognize that x^3 is the GCF of all of terms.
Notice that 3 is the smallest exponent
So, x^8 + 6x^5 - 11x^3 = x^3 (x^5 + 6x^2 - 11)

Notice that:
(x^3)(x^5) = x^8...just add the exponents
(x^3)(6x^2) = 6x^5...just add the exponents
(x^3)(11) = 11x^3
So, everything checks out nicely.

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It's no different with variable exponents.
Given m^(n-2) + 3m^(n) - 4m^(n+1)
Here, (n-2) is the smallest exponent [since n-2 is definitely less than n and n+1]
So, m^(n-2) is the GCF of all terms.

So, m^(n-2) + 3m^(n) - 4m^(n+1) = m^(n-2)[1 + 3m^2 - 4m^3]

Notice that:
m^(n-2)[1] = m^(n-2)
m^(n-2)[3m^2] = 3m^n ...just add the exponents
m^(n-2)[4m^3] = 4m^(n+1)...just add the exponents
So, everything checks out nicely.


I hope that helps.

Cheers,
Brent

[alexander.vien]

Thanks Brent! Much appreciated.

I was getting confused on the way to add the exponents because I always thought the bases needed to be the same. But your examples helped be understand that the coefficient near the variable can be separated with multiplication.

3m^n = 3 * m^n

(m^n-2)(3m^2) = (m^n-2 * m^2) * 3 = 3m^n