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## Algebra 2

### Unit 6: Lesson 2

Properties of exponents (rational exponents)

# Rewriting mixed radical and exponential expressions

Sal rewrites (r^(2/3)s^3)^2*√(20r^4s^5), once as an exponential expression and once as a radical expression. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• Would (2sqrt(5))(s^(17/2))(r^(10/3)) also be an acceptable answer?
• Yes, that is the same as his first answer, just with cleaner/more consistent exponent formatting.

I also wen't for the improper fractions when I did it as well. Just looks better to me and took a lot less work to stick with fractions the whole way through.
• Why does he take the 20 and factor it into 4 and 5 at , likewise factoring of the s ?
• The reason he factored the 20 into 4 and 5 was to simply the terms under the radical sign. Since 20 is not a perfect square, it is composed of a perfect square (the 4) multiplied by another number (the 5). Since we are trying to simplify this expression, our goal is to break down the 20 into simpler components. When we break the 20 down to 4 times 5, we are able to take the square root of the 4, simplifying the term. Hope that makes sense.
• what if i wrote the r as r^10/3?
• Yes, that would be fine. And actually, your way would be more conventional, as mathematicians tend to try to avoid mixed numbers because of the risk of confusion -- is 3 1/3 "three and a third", or "three multiplied by 1/3"? So I'd usually write it your way, as r^(10/3).
• At , why does Sal further break down the second segment of the equation with the square root under it? Can't you just for example, keep 20 and just raise it to the 1/2 power instead of doing all of that extra stuff Sal does? Thanks, and any help is appreciated.
• When it tells you to simplify something such as this, you want to simplify it as much as you can, by breaking it down into prime factors. That way, you can easily eliminate like factors. Hope that helps!
• is (r^2/3)^2 not r^4/9 ?
he sets it equivalent to r^4/3 at apx
• You have to be real careful here: if you had a regular fraction squared, e.g. 2/3, then sure you'd do (2/3)*(2/3). But here, the fraction is not a base you have to raise to a power; it is actually an exponent! That means that what you have to square is not 2/3, it's the whole (r^(2/3)). And the way you do that is by multiplying the two exponents: multiply 2/3 by 2, don't square it!

So basically, what Andrew said: you'd square 2/3 if you had r^((2/3)^2). And like he said, be very careful with how you write exponents and fractions. You have to use parentheses whenever needed. 4/8^3 is not 4/8/*4/8/*4/8 ; it's actually 4/(8*8*8). You should have written (4/8)^3. To be honest, I hadn't even paid attention to that in your first message, but this is the kind of mistakes that can really make your calculations a mess.
• Please! I'm in need of help with a problem: Simplify the expression;

(2x^3-1)^3 (4/3) (x^3-4)^1/3 (3x^2) + (x^3-4)^4/3 (3) (2x3-1)^2 (6x^2)
• Why don't we just use a calculator instead of dealing with this mess?
• Learning to do it by hand helps you get overall smart, and helps your brain make connections easier. When you go onto other units you may be able to use a calc. but think of this as good practice for SATs which make you do some math w/o a calculator
• how would you simplify Square root of z over z^1/3
• sqrt(z) is the same as z^(1/2)
So, to simplify z^(1/2) / z^(1/3), you use the exponent property for division and subtract the exponents. Sin the exponents are fractions, convert them to a common denominator to do the subtraction.
z^(1/2-1/3) = z^(3/6-2/6) = z^1/6

Hope this helps.
• At about , why does he use a fraction and a number for r?
• When you multiply same bases, you add exponents, so 4/3 + (4)1/2 = 10/3 which is an improper fraction, but to make it a proper fraction, we get 3 1/3. So 3 sets of 3 rs come out of the cubed root and one r stays in which he does not get to until the very end of the video. He does the same thing with s when he gets 8.5 which is the same as 8 1/2.
• How do you solve a question like 12sqrt(X)/4X^3?
• Heads up, LaTeX formatting is not available for Khan Academy. Using regular fraction syntax will suffice!

(12/sqrt(x))/4x³

Is simply saying:
12/sqrt(x) * 4x³ = 48x³/sqrt(x)

Using the fraction exponents property, we can rewrite this expression as:

48x³/x^(1/2)

Now using the dividing exponents property:

48 * x^(3-1/2) = 48x^(5/2) = 48sqrt(x⁵) = 48x²sqrt(x)

Hopefully that helps!