Simplify the exponential expression (-3x^2 y^4)^5.

Step-by-step solution

1. 1

   Apply the power of a product rule. $(abc)^n=a^nb^nc^n$, so every factor inside the parentheses — the coefficient included — gets the exponent $5$: $\left(-3x^2y^4\right)^5=(-3)^5\left(x^2\right)^5\left(y^4\right)^5.$

2. 2

   Raise the coefficient carefully. $(-3)^5=-3\cdot-3\cdot-3\cdot-3\cdot-3=-243$. An odd exponent keeps the sign negative, and the magnitude is $3^5=243$ — this is where the common answer $-27$ (which is only $(-3)^3$) goes wrong.

3. 3

   Apply the power of a power rule to the variables. Exponents multiply: $\left(x^2\right)^5=x^{10}$ and $\left(y^4\right)^5=y^{20}$.

4. 4

   Assemble the result. $\left(-3x^2y^4\right)^5=-243x^{10}y^{20}.$

5. 5

   Sanity-check with a value. Put $x=y=1$: the left side is $(-3)^5=-243$ and the right side is $-243$. The frequently seen $-27x^{10}y^{20}$ fails this test immediately.