Which is the simplified form of the expression [tex][tex]$\left(\left(p^2\right)\left(q^5\right)\right)^{-4} \cdot\left(\left(p^{-4}\right)\left(q^5\right)\right)^{-2} ?$[/tex][/tex]?

A. [tex][tex]$\frac{1}{9^{30}}$[/tex][/tex]
B. [tex][tex]$9^{30}$[/tex][/tex]
C. [tex][tex]$\frac{p^{12}}{q^{60}}$[/tex][/tex]
D. [tex][tex]$p^{12} q^{60}$[/tex][/tex]

Answer :

To simplify the expression [tex]\(\left(\left(p^2\right)\left(q^5\right)\right)^{-4} \cdot \left(\left(p^{-4}\right)\left(q^5\right)\right)^{-2}\)[/tex], we will use the properties of exponents step by step.

1. Start with the first part of the expression: [tex]\(\left(p^2 q^5\right)^{-4}\)[/tex].

Using the property of exponents [tex]\((a \cdot b)^n = a^n \cdot b^n\)[/tex], we get:
[tex]\[ (p^2 q^5)^{-4} = (p^2)^{-4} \cdot (q^5)^{-4}. \][/tex]

Simplify each term:
[tex]\[ (p^2)^{-4} = p^{2 \cdot (-4)} = p^{-8}, \][/tex]
[tex]\[ (q^5)^{-4} = q^{5 \cdot (-4)} = q^{-20}. \][/tex]

[tex]\[ (p^2 q^5)^{-4} = p^{-8} \cdot q^{-20}. \][/tex]

2. Next, consider the second part of the expression: [tex]\(\left(p^{-4} q^5\right)^{-2}\)[/tex].

Again, using the property of exponents [tex]\((a \cdot b)^n = a^n \cdot b^n\)[/tex]:
[tex]\[ (p^{-4} q^5)^{-2} = (p^{-4})^{-2} \cdot (q^5)^{-2}. \][/tex]

Simplify each term:
[tex]\[ (p^{-4})^{-2} = p^{-4 \cdot (-2)} = p^{8}, \][/tex]
[tex]\[ (q^5)^{-2} = q^{5 \cdot (-2)} = q^{-10}. \][/tex]

[tex]\[ (p^{-4} q^5)^{-2} = p^{8} \cdot q^{-10}. \][/tex]

3. Now, combine the two parts:

[tex]\[ \left(p^2 q^5\right)^{-4} \cdot \left(p^{-4} q^5\right)^{-2} = (p^{-8} \cdot q^{-20}) \cdot (p^8 \cdot q^{-10}). \][/tex]

Using the property of the product of powers with the same base, [tex]\((a^m \cdot a^n = a^{m+n})\)[/tex]:

Combine the [tex]\(p\)[/tex] terms:
[tex]\[ p^{-8} \cdot p^{8} = p^{-8 + 8} = p^{0} = 1. \][/tex]

Combine the [tex]\(q\)[/tex] terms:
[tex]\[ q^{-20} \cdot q^{-10} = q^{-20 + (-10)} = q^{-30}. \][/tex]

Therefore, the expression simplifies to:
[tex]\[ 1 \cdot q^{-30} = q^{-30} = \frac{1}{q^{30}}. \][/tex]

Thus, the simplified form of the given expression is:
[tex]\[ \frac{1}{q^{30}}. \][/tex]

None of the given choices directly match this result. Recheck if supplied choices or verify no particular isolated variable 'q' referring there. However, while interpreting near highest simplification under 'exponent context', closest re-depiction might allow similarly, but otherwise retouched choices given initial forms.

Typically matching steps realises scrutinize directly otherwise more consistently probable anticipated reframing choice should carefully reanalysed otherwise mostly clearest similar simplest validated depiction showing closest conversion correction typically reflecting mathematically [tex]\(\boxed{\frac{1}{q^{30}}}\)[/tex] .

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