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Algebra 2
Course: Algebra 2 > Unit 6
Lesson 3: Evaluating exponents & radicals- Evaluating fractional exponents
- Evaluating fractional exponents: negative unit-fraction
- Evaluating fractional exponents: fractional base
- Evaluating quotient of fractional exponents
- Evaluating mixed radicals and exponents
- Evaluate radical expressions challenge
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Evaluating fractional exponents: fractional base
CCSS.Math: ,
Sal shows how to evaluate (25/9)^(1/2) and (81/256)^(-1/4). Created by Sal Khan.
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At1:38Sal does the reciprocal to make the fraction positive, is this the same as moving the entire fraction to the denominator and going from there? Thanks!(12 votes)
It does end up with the same result to take the fraction and put it as a denominator under a one as the numerator, however it is faster and less confusing to do it the way Sal does in the example.(6 votes)
If I a number does not pop out at me when I am trying to simplify a radical, is there a quicker way to figure out its answer besides guessing?(6 votes)
That is a very good question. When simplifying a radical, the quickest method (besides intuition) is to make a factor tree. A factor tree is something you may have learned about in grade school. Here is a quick refresher:https://www.khanacademy.org/math/in-sixth-grade-math/playing-numbers/prime-factorization/v/prime-factorization
The next step would be to try and find number combinations that go in the same amount of times as the index. This video explains how to do that:https://www.khanacademy.org/math/algebra/rational-exponents-and-radicals/introduction-to-rational-exponents-and-radicals/v/radical-expressions-with-higher-roots
I hope this helps! Good luck!(15 votes)
- At0:45, wouldn't the 5/3 raised to the 2 power be written like this (5/3)^2 rather than like 5^2/3? Shouldn't the parentheses be there?(7 votes)
Yes, it should be in parentheses. He said the correct thing: "5/3 squared" but he sashayed on to the next example without putting in that clarifying set of parentheses.(5 votes)
Couldn't you also find the prime factorization to find 256^1/4 or 81^1/4?(4 votes)
Yes, that is the way that you would do it without a calculator. You could also just remember 4^4 is 256 and 3^4 is 81.(5 votes)
- why are you only looking for the principle root?(3 votes)
The negative roots are unnecessary for videos on negative fractional exponents.(1 vote)
What if you had a decimal number to the power of a negative exponent? for example 0.3 ^ -4(2 votes)
I don't know if you are familiar with the exponent rules. We can use the negative and power exponent rules:
Negative exponent: a^-1 = 1/ (a^1)
Power rule: (a^m)^n = a^(mn)
we can rewrite 0.3^-4 as 0.3^ (-1 * 4)
Using power rule we get (0.3^4)^-1
Using negative exponent rule we get 1/(0.3^4)
Which equals to 1/ (0.3 * 0.3 * 0.3 * 0.3) = 1/0.0081(2 votes)
How would I simplify a problem that is to the fourth root, but has a negative base? Something like (-625)^(3/4).(2 votes)
First things first, depends on the expression. You would either get a solution with i, or you'd get a rational number. Let's take your example:
(-625)^(3/4)
Best way to approach this is to split it up, remember that:
ab² = a²b² (² being an example)
-1^(3/4) * 625^(3/4)
Looking at 625, it can be broken down into:
5⁴
Using (aᵇ)ᶜ = aᵇᶜ:
((5)⁴)^(3/4) = 5³ = 125
Your answer would be:
125 * -1^(3/4)
Which cannot be simplified further !
Note that other cases might cannot simplify like this, it's simply a matter of breaking down the question.(2 votes)
- if it the problem is (125/27) ^ -2/3 how would you solve it?(2 votes)
(125/27)^(-2/3)=1/(125/27)^(2/3)=1/[(125/27)^(1/3)]^2=1/[(5/3)^2]=1/(25/9)=9/25.(2 votes)
At1:30Sal gets rid of the (-) sign by taking the reciprocal of the fraction, what I did was just divide the entire thing by 1, and then at the end I had 1/(3/4) which works out to 4/3 which is the right answer, but is there any particular reason why he chose to take the reciprocal instead?(1 vote)
You're lucky that you got 4/3. But what you did disregards what is actually happening mathematically. Dividing by 1 won't work on everything.
The fraction is 3/4 when it's reduced right? So the original statement was (3/4) ^ (-1/4) power. As you know, any number to any fractional power makes it the root of it.
Like a ^ (1/2) is the sqrt(a), right? Thus, a ^ (1/3) would be the CUBE root of a.
When we have a negative fraction as the power, we take the reciprocal because its the reversal of it. So a ^ (-1/2), would be 1 / sqrt(a). That means a/b ^ (-1/2) would be sqrt(b/a).
The only reason why (256/81)^(1/4) equaled 4/3 was because it was taking the fourth root of 256 and 81, which so happens to equal 4/3, or 1 DIVIDED by (81/256), which is what you did.
If he used a different fractional power, your answer would be wrong.(4 votes)
- how do you answer the question when the 4th root of a number doesn't work out so squeaky clean. e.g., 81 1/4th power through prime factorization works out clean with 3, 4's. Seeing as how I don't know how to find advanced numbers 3rd 4th 5th roots etc. other than by a prime factorization how do I represent it if i end up with something that doesn't have the exact number of prime factors as the root that I am taking it too?(2 votes)
- You can only simplify the root, which is not always easy nor does it leave a squeaky clean solution...but it does increase the possiblity of combining like terms if you have multiple terms in your expression or equation(2 votes)
1:38Video transcript
Fractional exponents can be
a little daunting at first, so it never hurts to do as
many examples as possible. So let's do a few. What if we had
25/9, and we wanted to raise it to the 1/2 power? So we're essentially
just saying, well, what is the principal
square root of 25/9? So what number times
itself is going to be 25/9? Well, we know 5 times 5
is 25, and 3 times 3 is 9. So why don't we
just go with 5/3? Because notice, if you
have 5/3 times 5/3, that is going to be 25/9. Or another way of saying
this, that 5/3 squared is equal to 25/9. So 25/9 to the 1/2 is
going to be equal to 5/3. Now let's escalate
things a little bit. Let's take a really hairy one. Let's raise 81/256 to
the negative 1/4 power. I encourage you to pause this
and try this on your own. So what's going on here? This negative-- the first
thing I always like to do is I want to get rid of this
negative in the exponent. So let me just take
the reciprocal of this and raise it to the positive. So I could just say that
this is equal to 256/81 to the 1/4 power. And so now I can say,
well, what number times itself times
itself times itself is going to be equal
to 256, and what number times itself times
itself times itself-- did I say that four times? Well, what number, if I take
four of them and multiply, do I get 81? And one way to
think about it, this is going to be the
same thing-- and we'll talk about this in
more depth later on when we talk about
exponent properties. But this is going to be
the exact same thing as 256 to the 1/4 over 81 to the 1/4. You, in fact, saw it over here. This over here was the same
thing as the square root of 25 over the square root of 9. Or 25 to the 1/2
over 9 to the 1/2. So we're just doing
that over here. So one, we still have to think
about what number this is. And this is a little bit of,
there's no easy way to do this. You kind of have
to just play around a little bit to come up with it. But 4 might jump out at you if
you recognize that 16 times 16 is 256. We know that 4 to
the fourth power, or you're about to know this,
is 4 times 4 times 4 times 4. And 4 times 4 is 16, times 4
is 64, times 4 is equal to 256. So 4 to the fourth
is 256, or we could say 4 is equal to
256 to the 1/4 power. Fair enough? Now what about 81? Well, 3 might jump out at you. We know that 3 to
the fourth power is equal to 3 times 3 times 3
times 3, which is equal to 81. So 3 is equal to 81 to the 1/4. So this top number, 256
to the 1/4, that's just 4. 81 to the 1/4, that is just 3. So this right over here is
going to be equal to 4/3.