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Question
Assuming that x, y, z are positive real numbers, simplify the following:
`(x^((-2)/3)y^((-1)/2))^2`
Solution
We have to simplify the following, assuming that x, y, z are positive real numbers
Given `(x^((-2)/3)y^((-1)/2))^2`
As x and y are positive real numbers then we have
`(x^((-2)/3)y^((-1)/2))^2=(x^((-2)/3)xxx^((-2)/3)xxy^((-1)/2)xxy^((-1)/2))`
By using law of rational exponents `a^-n=1/a^n` we have
`(x^((-2)/3)y^((-1)/2))^2=1/x^(2/3)xx1/x^(2/3)xx1/y^(1/2)xx1/y^(1/2)`
`(x^((-2)/3)y^((-1)/2))^2=1/(x^(2/3)xx x^(2/3))xx1/(y^(1/2)xxy^(1/2))`
By using law of rational exponents `a^m xx a^n=a^(m+n)` we have
`(x^((-2)/3)y^((-1)/2))^2=1/x^(2/3+2/3)xx1/y^(1/2+1/2)`
`=1/x^(4/3)xx1/y^(2/2)`
`=1/x^(4/3)xx1/y`
`=1/(x^(4/3)y)`
Hence the simplified value of `(x^((-2)/3)y^((-1)/2))^2` is `1/(x^(4/3)y)`
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