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Comment on Powers of 2
Is it mathematically correct
You could.
You could.
(2^-13 - 2^-9) = 2^-9(2^-4 - 1)
(2^-13 - 2^-17) = 2^-13(1 - 2^-4)
Divide to get (2^4)(-1)
Evaluate: -16
Thank you!
Awesome!!Thanks for your
I don't understand the
Let's start with some
Let's start with some straightforward factoring like these 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.
Let's factor the numerator: 2^(-13) - 2^(-9)
The two exponents are -13 and -9
Since -13 is the smaller exponent, we'll factor out 2^(-13)
So, we get: 2^(-13) - 2^(-9) = [2^(-13)][something goes here]
Now we need to determine what goes in the brackets.The first part is easy...
2^(-13) - 2^(-9) = [2^(-13)][1 - something]
What is "something" going to be?
We know that [2^(-13)] x [something] = 2^(-9)
When we multiply powers with the same bases, we ADD the exponents. So, we need -13 + ?? = -9
Well, -13 + 4 = -9, so the missing exponent is 4
So, we get: 2^(-13) - 2^(-9) = [2^(-13)][1 - 2^4]
That's the numerator.
We can use the same logic with the denominator to determine that...
2^(-13) - 2^(-17) = [2^(-17)][2^4 - 1]
So, our fraction can be rewritten as: [2^(-13)][1 - 2^4]/[2^(-17)][2^4 - 1]
Now we'll simplify this fraction in PARTS
First, [2^(-13)]/[2^(-17)] = 2^4 (we use the quotient law and subtract -17 from -13 to get the exponent 4)
Next, we need to recognize that [1 - 2^4]/[2^4 - 1] = -1
ASIDE: The property I used is: (x-y)/(y-x) = -1
So, [2^(-13)][1 - 2^4]/[2^(-17)][2^4 - 1] = (2^4)(-1)
Does that help?
Would it be a safe approach
Great strategy!!
Great strategy!!
We know that, since the numerator is negative and the denominator is positive, the correct answer must be NEGATIVE.
So, the correct answer is either A or B.
Also, the MAGNITUDE of the numerator is greater than the MAGNITUDE of the denominator, which means the value (after the negative sign) must be greater than 1.
Answer: A
Cheers,
Brent
What if the denominator was
Yes, the answer would still
Yes, the answer would still be guaranteed to be negative.
negative/positive = negative, and positive/negative = negative
Can we factor 2^-13 from the
You might want to read my
You might want to read my response to Dawnhughan above.
When we perform greatest common factor factoring, we need to find the greatest exponent that each term shares.
For example, let's factor x^7 + x^4 + x^3
In this case, x^3 is the greatest common factor of all three terms.
So, we'll factor it out to get:
x^7 + x^4 + x^3 = x^3(x^4 + x + 1)
Notice that we COULD factor out x^7 as well, but the results are not fun to work with.
x^7 + x^4 + x^3 = x^7[1 + x^(-3) + x^(-4)]
Likewise, if we factor 2^(-13) from the numerator and denominator, we get:
NUMERATOR: 2^(-13)[1 - 2^4]
DENOMINATOR: 2^(-13)[1 - 2^(-4)]
So, our fraction becomes: 2^(-13)[1 - 2^4]/2^(-13)[1 - 2^(-4)]
Simplify to get: [1 - 2^4]/[1 - 2^(-4)]
At this point, our fraction WILL simplify to be -16, but it will take more work to get there than had we factored out the greatest common factor each time.
[1 - 2^4]/[1 - 2^(-4)] = [1 - 16]/[1 - 1/16]
= [-15]/[15/16]
= [-15][16/15]
= -16
Cheers,
Brent
I just flipped the top and
I did 2^13-2^17/2^13-2^9
That strategy works, BUT for
That strategy works, BUT for one very specific reason:
It relies on the following exponent property: [b^(-x)]/[b^(-y)] = (b^y)/(b^x)
So, your approach works (this time!) because we can factor out negative powers in the numerator and denominator and then apply the above property.
Cheers,
Brent
if take 2 POW 13 in both
Sorry, I didn't follow that.
Sorry, I didn't follow that.
In order for the given fraction to equal 1, the numerator and denominator must be equal.
Can you elaborate on your approach?
Sorry my bad. Later I
Bering said that, what I meant is if I take 2^13 from both numerator and denominator. I got a solution equals 1.
But that's wrong. It was my calculation mistake..
Thanks for reply and follow up. Your course content and explanation for each topic is great!! :)
Thanks, I'm glad you like the
Thanks, I'm glad you like the course!