Rational Exponent

ⁿ√a^m = a^m/n

16 = 2⁴

Power

³√2⁴ = 2^4/3

So 2^4/3 is the answer.

a^7/9 = ⁹√a⁷

So ⁹√a⁷ is the answer.

4^5/2
= (2²)^5/2
= 2^2⋅[5/2]

Power of a Power

2⋅[5/2] = 5

2⁵ = 32

So 32 is the answer.

3^-7/2⋅3^5/2 = 3^{-7/2 + 5/2}

Product of Powers

-7/2 + 5/2 = -2/2

-2/2 = -1

3^-1 = 1/3

Negative Exponent

So 1/3 is the answer.

Change each radical
to a rational exponent power.

³√x² = x^2/3

⁶√x
= ⁶√x¹
= x^1/6

√x
= ²√x¹
= x^1/2

So
[³√x²⋅⁶√x]/√x
= [x^2/3⋅x^1/6]/x^1/2.

[x^2/3⋅x^1/6]/x^1/2
= x^{2/3 + 1/6 - 1/2}

Change the denominators to 6.

[2/3]⋅[2/2] = 4/6
-[1/2]⋅[3/3] = -3/6

4/6 + 1/6 - 3/6
= 5/6 - 3/6
= 2/6

2/6 = 1/3

x^1/3
= ³√x¹
= ³√x

So ³√x is the answer.