Question

A student was asked to simplify this expression
A) What error(s) did the student make?
B) What is the correct simplified form of the expression?

Answer 1

Let me solve

$\\ \rm\rightarrowtail \dfrac{-21x^5y^2}{7x^4y^5}$

$\\ \rm\rightarrowtail -3x^{5-4}y^{2-5}$

$\\ \rm\rightarrowtail -3xy^{-3}$

$\\ \rm\rightarrowtail \dfrac{-3x}{y^3}$

Answer 2

Answer:

$\dfrac{-21x^5y^2}{7x^4y^5}=\dfrac{-21}{7} \cdot \dfrac{x^5}{x^4} \cdot \dfrac{y^2}{y^5}$

Carry out the division of the coefficients:

$\implies -3 \cdot \dfrac{x^5}{x^4} \cdot \dfrac{y^2}{y^5}$

Apply exponent division rule  $\dfrac{a^b}{a^c}=a^{b-c}$

$\implies-3 \cdot x^{5-4} \cdot y^{2-5}$

$\implies-3 \cdot x^1 \cdot y^{-3}$

$\implies-3x y^{-3}$

(a)  The student did not divide the coefficients correctly.

$\dfrac{-21}{7}=-3$  NOT  $\dfrac13$

Also, they did not apply the exponent division rule correctly to the $x$ variable.  Instead of subtracting 4 from 5 → a positive exponent, they subtracted 5 from 4 → a negative exponent.

(b) $-3x y^{-3} \textsf{ or }-\dfrac{3x}{y^3} \textsf{ in rational form}$