Simplify ((x^8y^-4)/(16y^(4/3)))^(-1/4)

Problem

((x8*y(−4))/(16*y(4/3)))(−1/4)

Solution

1. Simplify the interior by combining the y terms in the denominator using the rule (ya)/(yb)=y(a−b) or y(−a)=1/(ya)

(x8*y(−4))/(16*y(4/3))=(x8)/(16*y(4/3)*y4)

(x8)/(16*y(4/3)*y4)=(x8)/(16*y(16/3))

2. Apply the negative exponent by taking the reciprocal of the fraction to change the sign of the outer exponent.

((x8)/(16*y(16/3)))(−1/4)=((16*y(16/3))/(x8))(1/4)

3. Distribute the exponent to each factor in the numerator and the denominator using the power of a power rule (am)n=a(m⋅n)

(16(1/4)*(y(16/3))(1/4))/((x8)(1/4))

4. Evaluate the constants and simplify exponents by calculating the fourth root of 16 and multiplying the fractional exponents.

16(1/4)=2

(y(16/3))(1/4)=y((16/3)⋅(1/4))=y(4/3)

(x8)(1/4)=x(8⋅(1/4))=x2

Final Answer

((x8*y(−4))/(16*y(4/3)))(−1/4)=(2*y(4/3))/(x2)

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