Simplifying a rational expression with exponent rules

Key takeaway: This problem tests how to simplify a quotient of monomials by first expanding powers, then canceling common factors correctly.

Key idea

To simplify a rational expression like

$\frac{-2x^2y^3}{6(xy^2)^3},$

you should first rewrite every power carefully before canceling anything. The main skill here is applying exponent rules to a product raised to a power:

$(xy^2)^3 = x^3y^6.$

That step is essential because it reveals all factors in the denominator.

Next, reduce the numerical coefficients and subtract exponents for like bases. This is not about canceling individual letters randomly; it is about dividing powers with the same base.

Step-by-step simplification

Start by expanding the denominator:

$6(xy^2)^3 = 6x^3y^6.$

So the expression becomes

$\frac{-2x^2y^3}{6x^3y^6}.$

Now simplify the coefficient:

$\frac{-2}{6} = -\frac13.$

Then subtract exponents for the common bases:

• For $x$: $x^2/x^3 = x^{2-3} = x^{-1} = \frac1x$
• For $y$: $y^3/y^6 = y^{3-6} = y^{-3} = \frac1{y^3}$

Putting everything together gives

$-\frac13\cdot \frac1x \cdot \frac1{y^3} = \frac{-1}{3xy^3}.$

So the correct choice is (d).

Common mistake to avoid

A frequent error is treating $(xy^2)^3$ as $x^3y^2$ instead of $x^3y^6$. The exponent 3 must apply to both factors inside the parentheses. Another mistake is canceling $x^2$ with $x^3$ as if the result were $x$, rather than remembering that division of like bases subtracts exponents. Also, the negative sign belongs with the whole numerator coefficient and should be carried through the simplification.

When you check your work, verify that the final answer has no negative exponents and no common factors left between numerator and denominator.

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Pitfalls the pros know 👇 The biggest trap is expanding the denominator incorrectly. In $(xy^2)^3$, both $x$ and $y^2$ get raised to the third power, so the correct expansion is $x^3y^6$, not $x^3y^2$. A second trap is simplifying only the coefficients and forgetting that powers of variables also reduce. If you divide $x^2$ by $x^3$, the result is $1/x$, not $x$. Keeping the exponent rule $a^m/a^n=a^{m-n}$ in mind prevents most of the mistakes on this type of question.

What if the problem changes? If the expression were

$\frac{-2x^2y^3}{6(xy)^3},$

the denominator would become $6x^3y^3$, and the simplified result would be

$\frac{-1}{3x y^0}= -\frac{1}{3x}.$

If instead the numerator had been $-12x^5y^8$, the same method would still apply: expand the denominator first, then divide coefficients and subtract exponents. The structure of the problem stays the same even when the powers change.

Tags: Exponent rules, Negative exponents, Rational expressions