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))`
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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))`
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