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

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!