Answer:
[tex]x^{-12}y^{2}z^{4}[/tex]
Step-by-step explanation:
We have to simplify the given expression given as
[tex][(x^{2}y^{3})^{-1}.(x^{-2}y^{2}z)^{2}]^{2}[/tex]
[tex][x^{-2}.y^{-3}.x^{-4}.y^{4}.z^{2}]^{2}[/tex]
[tex][x^{-2-4}.y^{-3+4}.z^{2}]^{2}=[x^{-6}.y^{1}.z^{2}]^{2}=x^{(-6-6)}.y^{1+1}.z^{2+2}=x^{-12}y^{2}z^{4}[/tex]
Therefore simplification of the given expression gives [tex]x^{-12}y^{2}z^{4}[/tex]
We want to simplify a given expression by using exponent properties.
We will get:
[tex][ (x^2*y^3)^{-1}*(x^{-2}*y^2*z)^2]^2 = \frac{z^4*y^2}{x^{12}}[/tex]
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The properties that we will use is:
[tex]a^n*a^m = a^{n + m}[/tex]
[tex](a^n)^m = a^{n*m}[/tex]
We start with the expression:
[tex][ (x^2*y^3)^{-1}*(x^{-2}*y^2*z)^2]^2[/tex]
Such that x, y, z ≠ 0
We specify this because we can't divide by zero, so these can't be zero.
First, we can use the second property to rewrite:
[tex][ (x^{-1*2}*y^{-1*3})*(x^{-2*2}*y^{2*2}*z^2)]^2 \\\[ (x^{-2}*y^{-3})*(x^{-4}*y^{4}*z^2)]^2[/tex]
Now we rewrite it to:
[tex][ (x^{-2}*y^{-3})*(x^{-4}*y^{4}*z^2)]^2\\ \\[ (x^{-2}*x^{-4})*(y^{-3}*y^{4})*z^2]^2[/tex]
Now we use the first property:
[tex][ (x^{-2}*x^{-4})*(y^{-3}*y^{4})*z^2]^2\\\[ (x^{-2-4})*(y^{-3+4})*z^2]^2 \\\[ (x^{-6})*(y)*z^2]^2[/tex]
Finally, we use the second property again:
[tex][ (x^{-6})*(y)*z^2]^2 \\[ (x^{-6*2})*(y^2)*z^{2*2}] \\[ (x^{-12})*(y^2)*z^{4}] \\\ $\frac{y^2*z^4}{x^{12}}[/tex]
Where in the last step I wrote the x part in the denominator because you wanted only positive exponents.
If you want to learn more, you can read:
https://brainly.com/question/3210664
