Simplify `(64x^2)^(-1/6)(32x^5)^(-2/5)` :

First use the power of a product rule: `(ab)^m=a^mb^m`

`64^(-1/6)(x^2)^(-1/6)(32)^(-2/5)(x^5)^(-2/5)`

Now use the power to a power rule: `(a^m)^n=a^(mn)`

`64^(-1/6)x^(-1/3)(32)^(-2/5)x^(-2)`

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`64^(-1/6)=1/(64^(1/6))=1/2` using the negative exponent rule. (Also, `64^(1/6)=root(6)(64)=2` )

`32^(-2/5)=1/(32^(2/5))=1/(2^2)=1/4` using the negative exponent rule and `32^(2/5)=(32^(1/5))^2=2^2=4`

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So, using the commutative property of multiplication,we have:

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Simplify `(64x^2)^(-1/6)(32x^5)^(-2/5)` :

First use the power of a product rule: `(ab)^m=a^mb^m`

`64^(-1/6)(x^2)^(-1/6)(32)^(-2/5)(x^5)^(-2/5)`

Now use the power to a power rule: `(a^m)^n=a^(mn)`

`64^(-1/6)x^(-1/3)(32)^(-2/5)x^(-2)`

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`64^(-1/6)=1/(64^(1/6))=1/2` using the negative exponent rule. (Also, `64^(1/6)=root(6)(64)=2` )

`32^(-2/5)=1/(32^(2/5))=1/(2^2)=1/4` using the negative exponent rule and `32^(2/5)=(32^(1/5))^2=2^2=4`

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So, using the commutative property of multiplication,we have:

`(1/2)(1/4)x^(-1/3)x^(-2)`

Use the product of powers rule: `a^m*a^n=a^(m+n)`

`1/8x^(-7/3)` Again using the negative exponent rule we have:

`1/(8x^(7/3))`

**The simplified form of `(64x^2)^(-1/6)(32x^5)^(-2/5)` is `1/(8x^(7/3))` **