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Comment on Powers of 4
I don't get how when you take
Great question! To answer it,
Great question! To answer it, consider these more straightforward factoring examples:
k^5 - k^3 = k^3(k^2 - 1)
m^19 - m^15 = m^15(m^4 - 1)
In the first example, we factored out k^3 since 3 is the smaller of the two exponents.
In the second example, we factored out m^15 since 15 is the smaller of the two exponents.
With 4^x and 4^(x-2), the exponents are x and (x-2). Here, the smaller exponent is (x-2), so we'll factor out the 4^(x-2) to get: 4^(x-2) times [something] = 4^x - 4^(x-2)
In this case, we must recognize that [4^(x-2)](4^2) = 4^x, since we must add the exponents, and (x-2) + 2 = x
Does that help?
Hi, Brent! Tried a different
LEFT
4^x - ( 4^x )(4^-2) divided by 5
4^x (1 - 1/4^2) divided by 5
4^x (15/16)(1/5)
RIGHT
= (4^2x)(4^1)
You're on the right track,
You're on the right track, but there's a problem with the right side.
It should be 4^(2x+1) - 4^(2x) = (4^2x)[4^1 - 1]
So, we get: (4^x)(15/16)(1/5) = (4^2x)[4^1 - 1]
Simplify: (4^x)(3/16) = (4^2x)[3]
Divide both sides by 3 to get: (4^x)(1/16) = 4^(2x)
Rewrite 1/16 to get: (4^x)[4^(-2)] = 4^(2x)
Simplify left side: 4^(x-2) = 4^(2x)
So, x-2 = 2x
Solve: x = -2
Wow!
Thanks so much. That gave me a sleepless night - lost a bit of confidence right there.
Thanks.
I don't understand how you
The key is that we always
The key is that we always factor out the term with the SMALLEST exponent.
For example, we take x^8 + x^5 + x^11 and factor out x^5.
We get: x^8 + x^5 + x^11 = x^5(x^3 + 1 + x^6)
With 4^x and 4^(x-2), the exponents are x and (x-2).
Here, the smaller exponent is (x-2), so we'll factor out the 4^(x-2)
At this point, the tricky part is determining what goes in the brackets.
When we multiply powers we ADD exponents.
So, [4^(x-2)][4^2] = 4^(x-2 + 2) = 4^x
So, 4^x - 4^(x-2) = [4^(x-2)](4^2 - 1)
Here are three more examples:
5^(x+1) - 5^(x+8) = 5^(x+1)[1 - 5^7]
2^(x+2) + 2^(x-1) = 2^(x-1)[2^3 + 1]
7^(x-3) + 7^(x-9) = 7^(x-9)[7^6 + 1]
Does that help?
Cheers,
Brent
HI Brent, I was also
5^(x+1) - 5^(x+8) = 5^(x+1)[1 + 5^(x+7)]
Shouldn't it be 5^(x+1) - 5^(x+8) = 5^(x+1)[1 - 5^(7)]?
I multiplied this to: [5^(x+1)] * [5^(x+7)] = 5^(2x+8) ?
You're absolutely right,
You're absolutely right, Kameron. Good catch!
I've edited my response above.
Hi Brent,
I solved the below question by factoring.
The value (2^-14 + 2^-15 + 2^-16 + 2^-17)/5 is how many times the value of 2^-17?
2^-14(1 + 2^-1 + 2^-2 + 2^-3)/5
2^-14 (1 + 1/2 + 1/4 + 1/8)/5
2^-14 (15/8)/5
2^-14 * 3/8
2^-14 * 3 * 2^-3
2^-17 * 3
So it is 3 times the value of 2^-17
Question link: https:/
Question link: https://gmatclub.com/forum/the-value-of-2-14-2-15-2-16-2-17-5-is-130682....
That approach works perfectly. Nice work!
Interesting. Although I did