Question 1

At 1:38 Sal 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!

Accepted Answer

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.

Question 2

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?

Accepted Answer

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!

Question 3

At 0:45, 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?

Accepted Answer

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.

Question 4

Couldn't you also find the prime factorization to find 256^1/4 or 81^1/4?

Accepted Answer

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.

Question 5

I can't find any videos on how to do 1/2 with an exponent of 2

Accepted Answer

Remember, an exponent tells you to multiply the base (1/2) by the number of times in the exponents.  
(1/2)^2 = (1/2)(1/2) = (1*1)/(2*2) = 1/4

Here's the video: https://www.khanacademy.org/math/cc-sixth-grade-math/x0267d782:cc-6th-exponents-and-order-of-operations/x0267d782:powers-of-fractions-and-decimals/v/powers-of-fractions

FYI - You can find topics by using the search bar at the top of any KhanAcademy page.  That's how I found the video.

Question 6

What if you had a decimal number to the power of a negative exponent? for example 0.3 ^ -4

Accepted Answer

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

Question 7

How would I simplify a problem that is to the fourth root, but has a negative base? Something like (-625)^(3/4).

Accepted Answer

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.

Question 8

if it the problem is (125/27) ^ -2/3 how would you solve it?

Accepted Answer

(125/27)^(-2/3)=1/(125/27)^(2/3)=1/[(125/27)^(1/3)]^2=1/[(5/3)^2]=1/(25/9)=9/25.

Question 9

why are you only looking for the principle root?

Accepted Answer

The negative roots are unnecessary for videos on negative fractional exponents.

Question 10

At 1:30 Sal 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?

Accepted Answer

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.

College Algebra

Course: College Algebra > Unit 13

Evaluating fractional exponents: fractional base

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Video transcript