Simplify (3x^3y^4)^2/(6x^5y^3)(x^3y^2)^4 1/x^7y^3 3/2x^11y^3 3x^11y^3/2 9x^6/y^3

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Simplify (3x^3y^4)^2/(6x^5y^3)(x^3y^2)^4 1/x^7y^3 3/2x^11y^3 3x^11y^3/2 9x^6/y^3

Algebra
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familiar with the exponent rules ?
Not much :/
alright exponent rules \[\huge\rm (ab)^m =a^mb^m\] numbers/variables in the parentheses raised by m power when we multiply same bases we should `add` exponents \[\huge\rm x^m \times x^n=x^{m+n}\] and when we divide same base , `subtract` their exponents \[\huge\rm \frac{ x^m }{ x^n }=x^{m-n}\]

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\[\huge\rm \frac{ \color{ReD}{(3x^3y^4)^2}}{(6x^5y^3)(x^3y^2)^4}\] start with first exponent rule i posted above
\[\huge\rm {(3x^3y^4)^2} = ??\]
Would it be (3x^5y^6)?
I'm not good at math :/
multiply the exponents
you will be one day
exponent rules \[\huge\rm (a^1b^1)^m =a^{1 \times m}b^{1 \times m}\] numbers/variables in the parentheses raised by m power according to this rule \[\huge\rm {(3x^3y^4)^2} = 3^3 x^{3 \times2}y^{4 \times3}\] every number/variable in the parentheses raised by 2 power multiply the exponents
3^3x^6y^12
sorry there is a typo \[\huge\rm {(3x^3y^4)^2} = 3^2 x^{3 \times2}y^{4 \times3}\] 3 to the 2 power not 3
Ok
the power tells us how many times we should multiply the base 3^2 = 3 times 3
Ok
ugh typo sorry there is a typo \[\huge\rm {(3x^3y^4)^2} = 3^2 x^{3 \times2}y^{4 \times2}\] 3 to the 2 power not 3 and y^4 times 2 not 3 now simplify that
\[(3x ^{3}y ^{4})^{2} = 3^{2}x ^{6}y ^{8}\]
yes right what about 3^2 = ?
9
yes right so \[\huge\rm \frac{ \color{ReD}{9x^6y^8}}{(6x^5y^3)(x^3y^2)^4}\] now apply the same exponent rule for (x^3y^2)^4
Would i combine them?
apply the exponent rule just like we did for the numerator
Because I got \[x ^{12}y ^{8}\]
would I multiply the 4 to the other one too?
no bec 4 is power of x^3 y^2 so that's it for this part \[\huge\rm \frac{ \color{black}{9x^6y^8}}{(6x^5y^3)x^{12}y^{8}}\] you can remove the parentheses from (6x^5y^3) bec there isn't any exponent outside the parentheses \[\huge\rm \frac{ \color{black}{9x^6y^8}}{6x^5y^3x^{12}y^{8}}\] now apply the 2nd exponent rule ~when we multiply same bases we should `add` exponents \[\huge\rm x^m \times x^n=x^{m+n}\]
\[6x ^{17}y ^{11}\]
nice \[\huge\rm \frac{ \color{black}{9x^6y^8}}{6x^{17}y^{11}}\] reduce the fraction 9/6 and apply the exponent rule when we divide same base , `subtract` their exponents \[\huge\rm \frac{ x^m }{ x^n }=x^{m-n}\]
3x^11y^3/2
hmm top exponent `minus` bottom exxponents so `6-17` = ?
oh -11 and -3
yes right now we need to change negative to positive exponent \[\huge\rm x^{-m}=\frac{ 1 }{ x^m }\] to convert negative to positive exponent u should flip the fraction , when you flip it the sign of the exponent would change
So 3/2x^11y^3
looks good
Thank you so much!
np good work you just need to practice more on this stuff then you will be an expert at exponent rules
I will keep practicing, thank you again.
o^_^o

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