Question

which expression is equivalent to \\(\frac{x^{2}y^{3}z^{2}}{xy^{-3}z^{2}}\\)? \\(\bigcirc\\) a. \\(x\\) \\(\bigcirc\\) b. \\(x^{3}z^{4}\\) \\(\bigcirc\\) c. \\(x^{2}y^{6}\\) \\(\bigcirc\\) d. \\(xy^{6}\\) \\(\bigcirc\\) e. \\(xy^{6}z\\)

Explanation:

Step1: Simplify \( x \)-terms

Use exponent rule \( \frac{a^m}{a^n}=a^{m - n} \). For \( x \): \( \frac{x^2}{x}=x^{2 - 1}=x^1 = x \).

Step2: Simplify \( y \)-terms

For \( y \): \( \frac{y^3}{y^{-3}}=y^{3-(-3)}=y^{3 + 3}=y^6 \).

Step3: Simplify \( z \)-terms

For \( z \): \( \frac{z^2}{z^2}=z^{2 - 2}=z^0 = 1 \) (we can ignore the factor of 1).

Step4: Combine results

Multiply the simplified \( x \), \( y \), and \( z \) terms: \( x\times y^6\times1 = xy^6 \).

Answer:

D. \( xy^6 \)