Lesson: Expanding Expressions

Comment on Expanding Expressions

Hi Brent.

In another video we could not combine 2^5 + 2^5 to equal 4^10. What we did was have: 2 * 2^5 = 2^6.

I was confused that -3a^2b - 3a^2b = -6a^2b.

Does this also mean that 2^5 + 2^5 = 4^5? That is clearly incorrect but what is the difference going on here?

gmat-admin's picture

You're referring to the expansion I perform at 5:45 in the above video.

It all comes down to the difference between the BASES (of powers) and COEFFICIENTS.

If we are adding powers, we cannot combine bases.
For example, 3² + 3² does not equal 6²
Likewise, x³ + x³ does not equal (2x)³

We can, however, combine the coefficients.
For example 5x³ + 3x³ = 8x³

IMPORTANT: With the term 5x³, we are not cubing 5x.
That is 5x³ does NOT equal (5x)³
Rather 5x³ = (5)(x³)

In other words, 5x³ = x³ + x³ + x³ + x³ + x³
Likewise, 2x³ = x³ + x³
So, 5x³ + 2x³ = (x³ + x³ + x³ + x³ + x³) + (x³ + x³) = 7x³

It's also important to point out that in the expression 3a^2b, the base is a, and the coefficient is 3.
That is, the base is NOT 3a.
If we wanted to say that the base is 3a, we'd write (3a)^2b

Since the coefficient is 3, we can write: 3a^2b = (3)(a^2b) = (a^2b) + (a^2b) + (a^2b)

So, 3a^2b + 3a^2b = (3)(a^2b) + (3)(a^2b)
= (a^2b) + (a^2b) + (a^2b) + (a^2b) + (a^2b) + (a^2b)
= 6(a^2b)
= 6a^2b

Likewise, -3a^2b - 3a^2b = -6a^2b

Does that help?

Here's the video that explains coefficients: https://www.gmatprepnow.com/module/gmat-algebra-and-equation-solving/vid...

Here's the video on simplifying expressions: https://www.gmatprepnow.com/module/gmat-algebra-and-equation-solving/vid...


I didn't get the solution of question 1 given after this video in reinforcement activities. please explain
gmat-admin's picture

Here's my full solution: https://gmatclub.com/forum/3-7-is-what-multiple-of-318364.html#p2474154

Is there a certain point in the solution you don't understand?
The main concept tested here is this: (x^a)(x^b) = x^(a+b)
So, for example: (2^3)(2^7) = 2^(3+7) = 2^10

The only difference with the linked question above is that some of the exponents or negative, but that doesn't change the fact that we must add the exponents when we are multiplying powers with the same base.

So, while expanding the given expression, our first task is to multiply 3^7 and 3^(-4).
Since both powers have the same base (3), we must add the exponents.
In other words: 3^7 x 3^(-4) = 3^[7+(-4)] = 3^3
Similarly, 3^7 x 3^(-5) = 3^[7+(-5)] = 3^2
And 3^7 x 3^(-6) = 3^[7+(-6)] = 3^1

And so on....

Does that help?


In this question, I tried to substitute the value of x by first using 1, and then -1 too. Unfortunately, it still didn't yield a single correct option..and the timer had already crossed 3 min.
I could have expanded it too though, I agree. But it just makes me question what my initial approach should be. Any tips?
gmat-admin's picture

Question link: https://gmatclub.com/forum/x-x-1-x-2-x-x-316333.html

I would say that, if the expression looks relatively easy to simplify, then go ahead and simplify it.
On the other hand, if you're not sure how to simplify the expression, the next best strategy is to test values.

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