Question

Simplify:

`(2x) ^{ (1)/(2) } * (2x ^(3) ) ^{ (3)/(2) }`
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Answer 1

`4x^5`

Explanation:

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

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

`= [(2x)^(1)/(2) * (2x)^(3)/(2)] * (x^2)^(3)/(2)`

`= (2x)^{(1)/(2) + (3)/(2)} * x^{{2} * (3)/(2)}`

`= (2x)^{(4)/(2)} * x^{(6)/(2)}`

`= 2^2x^2 * x^3`

`= 4x^5`

Answer 2

`\huge\boxed{(2x)^(1)/(2)*(2x^3)^(3)/(2)=4x^5}`

Explanation:

`(2x)^(1)/(2)*(2x^3)^(3)/(2)\qquad\text{use}\ (ab)^n=a^nb^m\\\\=2^(1)/(2)x^(1)/(2)*2^(3)/(2)(x^3)^(3)/(2)\qquad\text{use}\ (a^n)^m=a^(nm)\\\\=2^(1)/(2)x^(1)/(2)*2^(2)/(3)x^{(3)((3)/(2))}=2^(1)/(2)x^(1)/(2)*2^(2)/(3)x^(9)/(2)\\\\\text{use the commutative and associative property}\\\\=\left(2^(1)/(2)*2^(3)/(2)\right)\left(x^(1)/(2)* x^(9)/(2)\right)\qquad\text{use}\ a^n* a^m=a^(n+m)`

`=2^{(1)/(2)+(3)/(2)}x^{(1)/(2)+(9)/(2)}=2^(1+3)/(2)x^{(1+9)/(2)}=2^(4)/(2)x^(10)/(2)=2^2x^5=4x^5`