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Which expression is equivalent to [tex]\left(\frac{\left(2 a^{-3} b^4\right)^2}{\left(3 a^5 b\right)^{-2}}\right)^{-1}[/tex]? Assume [tex]a \neq 0[/tex], [tex]b \neq 0[/tex].

A. [tex]\frac{2}{3 a^4 b^{10}}[/tex]
B. [tex]\frac{4}{9 a^4 b^{10}}[/tex]
C. [tex]\frac{1}{36 a^4 b^{10}}[/tex]
D. [tex]\frac{36 a^4 b^{10}}{2}[/tex]



Answer :

To find an equivalent expression for [tex]\(\left(\frac{\left(2 a^{-3} b^4\right)^2}{\left(3 a^5 b\right)^{-2}}\right)^{-1}\)[/tex], let's proceed step-by-step to simplify the given expression.

1. Expand the numerator and denominator inside the fraction:

We start with the given expression:
[tex]\[ \left(\frac{\left(2 a^{-3} b^4\right)^2}{\left(3 a^5 b\right)^{-2}}\right)^{-1} \][/tex]

2. Simplify the numerator [tex]\((2 a^{-3} b^4)^2\)[/tex]:

[tex]\[ (2 a^{-3} b^4)^2 = 2^2 \cdot (a^{-3})^2 \cdot (b^4)^2 \][/tex]
[tex]\[ = 4 \cdot a^{-6} \cdot b^8 \][/tex]

3. Simplify the denominator [tex]\((3 a^5 b)^{-2}\)[/tex]:

[tex]\[ (3 a^5 b)^{-2} = 3^{-2} \cdot (a^5)^{-2} \cdot (b)^{-2} \][/tex]
[tex]\[ = \frac{1}{3^2} \cdot a^{-10} \cdot b^{-2} \][/tex]
[tex]\[ = \frac{1}{9} \cdot a^{-10} \cdot b^{-2} \][/tex]
[tex]\[ = \frac{a^{-10} b^{-2}}{9} \][/tex]

4. Combine the simplified forms of the numerator and denominator:

[tex]\[ \frac{4 a^{-6} b^8}{\frac{a^{-10} b^{-2}}{9}} = 4 a^{-6} b^8 \times \frac{9}{a^{-10} b^{-2}} \][/tex]

5. Simplify the multiplication:

[tex]\[ = 4 \times 9 \times a^{-6} \times a^{10} \times b^8 \times b^2 \][/tex]
[tex]\[ = 36 \times a^{-6 + 10} \times b^{8 + 2} \][/tex]
[tex]\[ = 36 \times a^4 \times b^{10} \][/tex]

6. Take the inverse (as indicated by the outer exponent [tex]\(-1\)[/tex]):

[tex]\[ \left(36 a^4 b^{10}\right)^{-1} = \frac{1}{36 a^4 b^{10}} \][/tex]

Therefore, the expression [tex]\(\left(\frac{\left(2 a^{-3} b^4\right)^2}{\left(3 a^5 b\right)^{-2}}\right)^{-1}\)[/tex] simplifies to:

[tex]\[ \boxed{\frac{1}{36 a^4 b^{10}}} \][/tex]