Radicals as Rational Exponents

Question

Use rational exponents to simplify. Write answers using radical notation and no fraction exponents. Assume all variables are positive.

( ⁴√3x⁵y⁴ )¹⁶

Answer 1

Apply

(A × B)^C = A^C × B^C

and

(A^C)^D = A^C×D

and

( ^C√A )^D = A^D/C

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( ⁴√3x⁵y⁴ )¹⁶

= ( ⁴√3 )¹⁶ × ( x⁵ )¹⁶ × ( y⁴ )¹⁶

= 3^16/4 × x^16×5 × y^16×4

= 3⁴ × x⁸⁰ × y⁶⁴

= 3⁴x⁸⁰y⁶⁴

= 81x⁸⁰y⁶⁴

If the problem was meant to be

( ⁴√(3x⁵y⁴ ))¹⁶

then this can be simplified to

= (3x⁵y⁴)^16/4

= (3x⁵y⁴)⁴

= 3⁴ × x²⁰ × y¹⁶

= 3⁴x²⁰y¹⁶

= 81x²⁰y¹⁶

Answer 2

( ⁴√3x⁵y⁴ )¹⁶ I am assuming that the 3x⁵y⁴ is ALL under the fourth root symbol.

So using fractional exponents, this becomes

(3^1/4 x^5/4 y^4/4)¹⁶ Raising a power to another power says multiply the exponent, so...

3^{(1/4 x 16)} x^{(5/4 x 16)} y^{(4/4 x 16)} = 3⁴ x²⁰ y¹⁶