Question

there are some algebraic expressions: ( \frac{x^{2}y^{3}}{z} ), ( \frac{b^{2}c^{3}}{a} ), ( x^{2}y^{3}z^{-4} ), ( \frac{x^{2}y^{3}}{z^{4}} ), ( x^{2}y^{3}z^{-4} ) (arranged in colored blocks).

Explanation:

Step1: Recall negative exponent rule

A term $z^{-n} = \frac{1}{z^n}$

Step2: Rewrite the given expression

$$x^2y^3z^{-4} = x^2y^3 \cdot \frac{1}{z^4} = \frac{x^2y^3}{z^4}$$

Step3: Match to equivalent expression

Compare with the options: $\frac{x^2y^3}{z^4}$ matches the orange box expression, and $x^2y^3z^{-4}$ matches the bottom blue box expression.

Answer:

The expressions equivalent to $x^2y^3z^{-4}$ are:

1. $x^2y^3z^{-4}$ (bottom blue box)
2. $\frac{x^2y^3}{z^4}$ (orange box)