```Question 175802
number 13:
{{{sqrt(45*x^23)}}}
to simplify you need to remove everything from under the radical as much as possible.
let's see how that would work in this case.
{{{sqrt(45*x^23) = sqrt(9*5*x^11*x^11*x) = sqrt(3^2*5*(x^11)^2*x) = 3*x^11*sqrt(5x)}}}.
how do i know i got the right answer?
i cheated by using the calculator.
i took a random value of x and solved for it in the original equation and in the simplified equation and got the same answer.
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how did i know to do what i did?
couple of concepts that helped:
{{{sqrt(x^2) = x}}}
{{{x^a*x^a = x^(2a)}}}
{{{x^a*x^b = x^(a+b)}}}
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these are all basic concepts that you should review if you're still confused by them.
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number 14:
{{{root(4,(x^5)/(16y^15))}}}
again, you need to simplify by removing everything from under the radical that can be removed.
after looking at this i came up on the following:
{{{root(4,(x^5)/(16y^15)) = root(4,(x^4*x)/(2^4*(y^3)^4*y^3)) = x/(2*y^3)*root(4,x/y^3)}}}.
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again i cheated to prove the answer was correct by solving the original equation and the simplified equation for random values of x and y.
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i used the same concepts, except in this case i used:
{{{root(4,x^4) = x}}}
i also looked at {{{y^15}}} to see how i could get something to the 4th power out of it so that i could exctract that as well.
since {{{y^12 = (y^3)^4}}} i figured that if i could extract {{{y^12}}} from {{{y^15}}}, then i could also take that from under the radical.
since {{{y^15 = y^12*y^3}}} i was able to do that.
in order to do that, the concept used was:
{{{x^a*x^b = x^(a+b)}}}
in this case:
{{{y^12*y^3 = y^(12+3) = y^15}}} so i was good to go.
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number 15:
5) Simplify (x1/4 y1/2)2(x2y3)1/2
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i'm not quite sure what you mean by this but i'll take a stab at it.
i assume you mean:
{{{(x^(1/4)*y^(1/2))^2*(x^2*y^3)^(1/2)}}}
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if i'm correct in my assumption then this is how you would solve this problem.
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one of the concepts used is {{{(x^a)^b = x^(a*b)}}}
another concept used is {{{(x^a*y^b)^c = x^(ac)*y^(bc)}}}
{{{(x^(1/4)*y^(1/2))^2 = (x^(1/4))^2*(y^(1/2))^2 = x^(1/2)*y}}}
that's the left hand part of this equation.
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{{{(x^2*y^3)^(1/2) = (x^2)^(1/2)*(y^3)^(1/2) = x*y^(3/2)}}}
that's the right hand part of this equation.
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putting these together, you get:
{{{x^(1/2)*y*x*y^(3/2))}}}
combining like terms using the concept {{{x^a*x^b = x^(a+b)}}}, we get:
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{{{x^(1/2)*y*x*y^(3/2) = x^(1+(1/2)) * y^(1+(3/2)) = x^(3/2)*y(5/2)}}}.
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i cheated again to prove the answer is correct.
looks like all 3 answers are good.
these are not simple problems and you have to keep track of everything as you work through them.
they all require the use of basic concepts to get the answers.
i would recommend you review the basic concepts again now that you have some idea of how they can be used.
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if you would like to do a review of the basic concepts, then i would recommend this website:
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click on tutorial 26 (exponents).
they give a pretty decent review that is fairly easy to understand.
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try also this website: