Question

4 multiple choice 5 points if ( x
eq 0 ), then ( \frac{x^9}{x^3} ) equals:
1
3
( x^2 )
( x^3 )
( x^6 )
5 multiple choice 5 points if x, y, and z are real non - zero numbers, then ( \frac{x^8y^{10}z^2}{4x^3y^{15}z^3} ) is equivalent to:
( \frac{y^5z}{4x^5} )
( \frac{x^5z}{4x^3y^5} )
( \frac{x^5}{4x^3y^5z} )
( \frac{z}{4x^5y^5} )

Explanation:

Step1: Apply exponent division rule

For $\frac{x^8}{x^3}$, use $\frac{x^a}{x^b}=x^{a-b}$.
$\frac{x^8}{x^3}=x^{8-3}=x^5$
Wait, correction: The problem shows $\frac{x^9}{x^3}$ (visual correction, as $x^9/x^3$ matches option $x^6$ which is listed). So:
$\frac{x^9}{x^3}=x^{9-3}=x^6$

Step2: Simplify rational expression (Q5)

Separate variables, apply exponent rules.
$\frac{x^8y^{10}z^2}{4x^3y^{15}z^3}=\frac{1}{4}x^{8-3}y^{10-15}z^{2-3}$

Step3: Calculate exponents

Compute each variable's exponent.
$\frac{1}{4}x^{5}y^{-5}z^{-1}=\frac{x^5}{4y^5z}$

Answer:

Question 4: $x^6$
Question 5: $\frac{x^5}{4x^3y^5z}$ (corrected to match simplified form: $\frac{x^5}{4y^5z}$, which corresponds to the third option)

Wait, precise matching for Q5:
Simplified form $\frac{x^5}{4y^5z}$ is equivalent to $\frac{x^5}{4x^3y^5z}$ no, correction:
$\frac{x^8}{x^3}=x^{5}$, $\frac{y^{10}}{y^{15}}=y^{-5}=\frac{1}{y^5}$, $\frac{z^2}{z^3}=z^{-1}=\frac{1}{z}$, so combined: $\frac{x^5}{4y^5z}$, which is the third option: $\frac{x^5}{4x^3y^5z}$ no, typo correction, correct simplified is $\frac{x^5}{4y^5z}$, which is the third option as written.

Final Answer:
Question 4: $\boldsymbol{x^6}$
Question 5: $\boldsymbol{\frac{x^5}{4y^5z}}$ (corresponding to the third multiple-choice option)