Question

Simplify each expression. Assume that all variables are positive.

1)
(16x5y10/81xy2)3/4

2)
(−64)−2/3

3)
a2/3×a1/2

Answer 1

Q1. The answer is
`(8x^(3)y^(6) )/(27)`


`( (16 x^(5) y^(10))/(81x y^(2) ) )^{ (3)/(4) }= ( (16)/(81)* ( x^(5) )/(x)* ( y^(10) )/(y^(2)) )^{ (3)/(4) } \\ \\ ( x^(a) )/( x^(b) )= x^(a-b) \\ \\ ( (16)/(81)* ( x^(5) )/(x)*( y^(10) )/(y^(2)) )^{ (3)/(4) }}=( (16)/(81 )* x^(5-1)* y^(10-2))^{ (3)/(4) }=( (16)/(81 )* x^(4)* y^(8))^{ (3)/(4) }= \\ \\ = ((16)/(18) )^{ (3)/(4) }*(x^(4))^{ (3)/(4) }*(y^(8))^{ (3)/(4) }=`

`\frac{(16)^{ (3)/(4) }}{(18)^{ (3)/(4) }}*(x^(4))^{ (3)/(4) }*(y^(8))^{ (3)/(4) }=\frac{( 2^(4) )^{ (3)/(4) }}{( 3^(4) )^{ (3)/(4) }}*(x^(4))^{ (3)/(4) }*(y^(8))^{ (3)/(4) } \\ \\ (x^(a) )^(b) = x^(a*b) \\ \\ \frac{( 2^(4) )^{ (3)/(4) }}{( 3^(4) )^{ (3)/(4) }}*(x^(4))^{ (3)/(4) }*(y^(8))^{ (3)/(4) } = \frac{ 2^{4* (3)/(4) } }{ 3^{4* (3)/(4) } } * x^{4* (3)/(4) } * y^{8*(3)/(4)} = ( 2^(3) )/( 3^(3) ) * x^(3) *y^(6) =`

`= (8x^(3)y^(6) )/(27)`

Q2. The answer is 1/16


`(-64) ^ (-2)/(3) =(-1* 2^(6) ) ^ (-2)/(3)=(-1)^ (-2)/(3) *(2^(6) ) ^ (-2)/(3) \\\\x^(-a) = (1)/( x^(a) ) \\\\(-1)^ (-2)/(3) *(2^(6) ) ^ (-2)/(3) = (1)/((-1)^ (2)/(3)) *(1)/((2^(6))^ (2)/(3)) \\ \\ (x^(a) )^(b)=x^(a*b) \\\\x^{ (a)/(b) = \sqrt[b]{ x^(a) } } \\ \\`

`(1)/((-1)^ (2)/(3)) *\frac{1}{2^{6*(2)/(3)}} = \frac{1}{ \sqrt[3]{(-1)^(2) } } * (1)/( 2^(4) ) = \frac{1}{ \sqrt[3]{1} } * (1)/(16) = (1)/(1) * (1)/(16)= (1)/(16)`


Q3. The answer is
`a^{ (7)/(6) }`


`a^{ (2)/(3) } * a^{ (1)/(2) } \\ \\ x^(a)* x^(b) =x^(a+b) \\ \\ a^{ (2)/(3) } * a^{ (1)/(2) }= a^{ (2)/(3) + (1)/(2) } =a^{ (2*2)/(3*2) + (1*3)/(2*3) }=a^{ (4)/(6) + (3)/(6) }=a^{ (4+3)/(6) }=a^{ (7)/(6) }`