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Course: Algebra 2>Unit 6

Lesson 3: Evaluating exponents & radicals

Evaluating quotient of fractional exponents

Sal simplifies the complicated expression 256^(4/7) / 2^(4/7)  until he finds that the expression is equal to 16.

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• I am not sure why one cant subtract the exponents while we are dividing. Given:
256^(4/7) / 2^(4/7) Why does the rule of subtracting exponents(4/7 - 4/7) in this situation fail? Thank you
• Your bases have to be the same for the rule of subtracting exponents.
• What about the exponent is a irrational number?Like we describe 2^0.5=sqrt(2) , but how do we define 2^sqrt(2)?
• Doesn't simplifying the fraction (dividing 256 by 2 to reach 128) violate the PEMDAS rule whereby operation of exponent precedes that of division or multiplication?
• There is a property of exponents that tells us that having a fraction raised to an exponent is the same as having both the numerator and denominator individually raised to the exponent.
For example: (1/2)^3 = 1^3/2^3
The problem in the video is both the numerator and denominator with the same exponent. So, Sal uses this property exponents to bring the fraction back together, which allows him to then do the division.
Hope this helps.
• So 3.14 = 3/4?
I Dont Understand
• I thought in the exponents rules, a^m / a^n would end up having the exponents minus each other out and in (256^4/7) / (2^4/7) would become 256^0 / 256^0 and therefore would equal 1/1 and then 1. How come it's not like this?
• All properties of exponents require that you have a common base. As you noted: a^m/a^n = a^(m-n), but the a's must be the same.

256 does not equal 2. So, there is no common base which is why your approach doesn't work.

Sal used the property of a^m/b^m = (a/b)^m. This let him divide 256/2, then apply the exponent to get to his answer of 16.

Alternatively, you can convert to a common base.
256 = 2^8
(2^8)^(4/7) = 2^(8*4/7) = 2^(32/7)
Then apply the property: a^m/a^n = a^(m-n)
2^(32/7) / 2^(4/7) = 2^(32/7-4/7) = 2^(28/7) = 2^4 or 16

Hope this helps.
• why do we need to know this when we could just use a calculator
• At , what dos he mean by computationally intensive?
• A computation is just a mathematical calculation, so to say something is computationally intensive meaning that its hard to calculate. For instance, 1+1 is very easy to compute, so easy in fact you probably did it in your head without even thinking. Something like 128^4, or 53619 x 79863 would be considered hard to compute (without a calculator of course (unless perhaps you're a genius))
• Can't you subtract exponents when dividing? Thank You.
• Yes. If you have 2 values with a common base, you can subtract their exponents to do the division.
For example: 2^(5/4) / 2^(1/2) = 2^(5/4-1/2)
Since you are subtracting fractions, you need a common denominator: 2^(5/4-2/4) = 2^(3/4)

Hope this helps.
• (128^4)^(1/7) is actually fun, I love doing things like this.

128^2 = 16384
128^3 = 2097152
128^4 = 268435456

(268435456)^1/7 or 7rt(268435456) = 16

To check this math, you can multiply 16 seven times and get the answer

16^1=16
16^2=256
16^3=4096
16^4=65536
16^5=1048576
16^6=16777216
16^7=268435456

I don't recommend doing it this way, however. It takes math to a new level of annoyance. I just did it to show you guys what he meant by computationally intensive.

I need to go outside... touch grass... and play in the sun for the first time.

Also apparently, the answer is wrong... I know it is 2, but things don't turn out like they should when higher numbers cause you to have brainfarts. :)