Question

Which of the following simplifies the expression shown below?

a) (−27x^6z^4/y^−3)^23
b) 27x^6z^4/y^3
c) (−27^23)x^6z^4/y^3
d) (−27x^6z^4)^23/y^3

Answer 1

The given expression (-27x^6z^4/y^{-3})^{2/3} simplifies to 9x^4z^(8/3)/y^2 by applying the fractional exponent to each factor inside the parentheses.

Step-by-step explanation:

The expression in question is (-27x^6z^4/y^{-3})^{2/3}. To simplify this, we need to apply the power to each factor inside the parentheses. According to the exponentiation rules, we distribute the fractional exponent across all factors inside the partition.

Therefore, simplifying we get:

• For -27: The cube root of -27 is -3, and then squaring that gives 9.
• For x^6: The cube root of x^6 is x^2, and then squaring that gives x^4.
• For z^4: The cube root of z^4 is z^(4/3), and then squaring that gives z^(8/3).
• For y^{-3}: We apply the negative exponent rule which flips y to the numerator and compute the fractional exponent similarly to the other variables: y^2.

The simplified expression is therefore 9x^4z^(8/3)/y^2.