Question

Which of the following is equal to [((x²y³)/(x⁶y³x))⁻²]⁻³?

(A) (x⁴y⁻⁶)⁻³
(B) (x⁴y⁻⁶)⁻³
(C) (x⁸y⁻⁶)⁻³
(D) (x⁻⁸y⁻⁶)⁻³
(E) None of the above

Answer 1

The expression simplifies to x⁻¹²y⁻⁹. Option (A) (x⁴y⁻⁶)⁻³, when expanded, matches the simplified form. Therefore, the correct answer is (A) (x⁴y⁻⁶)⁻³.

Step-by-step explanation:

The given expression is
`\[ \left[ \left( (x²y³)/(x⁶y³x) \right)^(-2) \right]^(-3) \].`

To simplify this expression, first simplify the innermost part, i.e., the term inside the innermost parentheses. Here,
`\[ (x²y³)/(x⁶y³x) \]`can be simplified to
`\[ (1)/(x⁴) \].`

Now, substitute this simplified expression back into the original expression:
`\[ \left( (1)/(x⁴) \right)^(-2) \]⁻³.`

Simplify further:
`\[ (x⁴)^(-3) \].`

Finally, simplify the outermost expression:
`\[ (x⁴)^(-3) = x^(-12) \].`

So, the final simplified expression is
`\[ x^(-12)y^(-9) \].`

Now, compare this with the options provided. Option (A) is
`\[ (x⁴y^(-6))^(-3) \]`, which simplifies to . Since the exponents match the simplified expression, the correct answer is (A)
`\[ (x⁴y^(-6))^(-3) \].`