# How do you simplify this exponential expression?The expression is: (64x^2)^-1/6(32x^5)-2/5

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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))` **

The expression `(64x^2)^(-1/6)*(32x^5)^(-2/5)` has to be simplified.

Use the relations:

`(x^a)^b = x^(a*b)`

`x^a*x^b = x^(a+b)`

`(x*y)^a = x^a*y^a`

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

= `(2^6*x^2)^(-1/6)*(2^5*x^5)^(-2/5)`

= `(2^6)^(-1/6)*(x^2)^(-1/6)*(2^5)^(-2/5)*(x^5)^(-2/5)`

= `2^(6*(-1/6))*x^(2*(-1/6))*2^(5*(-2/5))*x^(5*(-2/5))`

= `2^-1*x^(-1/3)*2^-2*x^-2 `

= `2^(-1 - 2)*x^(-1/3 - 2)`

= `2^-3*x^(-7/3)`

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