Question: Powers of 4

Comment on Powers of 4

Pls help me to understand how the above equation was factorized.
greenlight-admin's picture

ASIDE: A lot of students struggle to see how we can factor 4^x - 4^(x-2) to get 4^(x-2)[4^2 - 1]

Before we examine the above factorization, let's look at some straightforward factoring examples:
k^5 - k^3 = k^3(k^2 - 1)
m^19 - m^15 = m^15(m^4 - 1)

IMPORTANT: Notice that, each time, the greatest common factor of both terms is the term with the SMALLER EXPONENT.

So, in the expression 4^x - 4^(x-2), the two exponents are x and (x-2). The smaller exponent is (x-2), so we can factor out 4^(x-2)

Likewise, w^x + x^(x+5) = w^x(1 + w^5)
And 2^x - 2^(x-3) = 2^(x-3)[2^3 - 1]

Does that help?

I'm still not understanding how factoring out 4^(x-2) gets us 4^(x-2) * (4^2-1) in the numerator. Specifically, where is 4^2 coming from? How can 4 still be squared once you factor out 4^(x-2)?
greenlight-admin's picture

Let's look at a different example.
Let's say we want to factor x⁷ + 5x⁴ - x³
Since the smallest exponent is 3, we can factor out the x³ to get:
x⁷ + 5x⁴ - x³ = x³(something)

At this point our job is to determine which terms go in the brackets.
The key here is that when we multiply x³ by each term, the product must equal the original expression x⁷ + 5x⁴ - x³
So, for example, the first term must be x⁴, since (x³)(x⁴) = x⁷
The second term must be 5x, since (x³)(5x) = 5x⁴
The third term must be -1, since (x³)(-1) = -x³
----------------------------

The same applies to the expression 4^x - 4^(x-2)
Since the smallest exponent is (x-2), we can factor out the 4^(x-2) to get:
4^x - 4^(x-2) = 4^(x-2)[something]
Once again, our job is to determine which terms go in the brackets.
The first term must be 4^2 since [4^(x-2)][4^2] = 4^x (we add the exponents)
The second term must be -1 since [4^(x-2)][-1] = 4^(x-2)
So, 4^x - 4^(x-2) = 4^(x-2)[4^2 - 1]

Does that help?
If not, here's an article I wrote on this topic: https://www.reddit.com/r/GMAT/comments/eplygy/factoring_expressions_with...

Cheers,
Brent

I get how you came by the smaller exponent but not the remaining terms in parenthesis - 4^2 - 1
greenlight-admin's picture

A lot of students struggle to see how we can factor 4^x - 4^(x-2) to get 4^(x-2)[4^x - 1]

Before we look at that, however, let's look at some more straightforward examples:
k^5 - k^3 = k^3(k^2 - 1)
m^19 - m^15 = m^15(m^4 - 1)
x^6 - x^5 + x^2 = x^2(x^4 - x^3 + 1)

IMPORTANT: Notice that, each time, the greatest common factor of both terms is the term with the SMALLEST exponent.

So, in the expression 4^x - 4^(x-2), the term with the SMALLEST exponent is 4^(x-2), since (x-2) is less than x.

This means we can factor out 4^(x-2)

At this point, we need to figure out what we must multiply 4^(x-2) by to get the end result of 4^x - 4^(x-2)

That is, we want 4^(x-2)[what goes here?] = 4^x - 4^(x-2)

Let's start with the first term. What must we multiply 4^(x-2) by to get 4^x?

Well, when we multiply powers with the same base, we ADD the exponents. So, we need to find an exponent that, when added to (x-2), gives a sum of x.

Well, (x-2) + 2 = x

So, the missing exponent must be 2. That is, 4^(x-2) times 4^2 = 4^x

So, we have one of our missing pieces.

We have: 4^(x-2)[4^2 - something] = 4^x - 4^(x-2)

The next piece of the puzzle is easier. We know that 4^(x-2) times 1 equals 4^(x-2)

So, we get: 4^(x-2)[4^2 - 1] = 4^x - 4^(x-2)

Here are a few more factoring examples involving variables in the exponents:

w^x + x^(x+5) = w^x(1 + w^5)
2^x - 2^(x-2) = 2^(x-2)[2^2 - 1]
5^x - 5^(x-3) = 5^(x-3)[5^3 - 1]

i didn't get this one you put it in the example.
w^x + x^(x+5) = w^x(1 + w^5)
shouldn't the bases must be matched.
but here I can see x and w.
greenlight-admin's picture

That doesn't really apply when it comes to multiplying by 1
That said, we COULD write: w^x + x^(x+5) = w^x(w^0 + w^5) = w^x(1 + w^5)

Yes i got it. Thank you everyone

Hi Brent, how do we decide to factor out small or big exponent?
In the 2nd part, If I have factored out 4^2x+1 instead, I got 4^2x+1 (1 - 4)
4^2x+1 (-3). What went wrong here? Could you help clarify? Thanks
greenlight-admin's picture

Always factor out the smallest exponent.
For more on this, read my comment above from March 28, 2017

Get it thanks Brent

Practice Tests

Be sure to take some full-length practice tests as part of your preparation. This will strengthen your test-taking skills (time management, endurance,etc.), and it will help identify any remaining areas of weakness. 

Test Day Mindset

As part of your preparation, work on adopting the proper mindset/attitude on test day. This will do wonders for your score. 

Free “Question of the Day” emails!