Question

q4 multiply the rational expressions and simplify.
\\(\frac{3xz^2}{4xz} \cdot \frac{4x^2}{2z}\\)
\\(\bigcirc \frac{12x^3z^2}{8xz}\\)
\\(\bigcirc \frac{12x^3z^2}{8xz^2}\\)
\\(\bigcirc \frac{3x^2z}{2}\\)

Explanation:

Step1: Multiply numerators and denominators

To multiply the rational expressions \(\frac{3xz^2}{4xz} \cdot \frac{4x^2}{2z}\), we first multiply the numerators together and the denominators together.
The numerator of the product is \(3xz^2 \cdot 4x^2 = 12x^{1 + 2}z^2 = 12x^3z^2\) (using the rule of exponents \(a^m \cdot a^n = a^{m + n}\) for \(x\)).
The denominator of the product is \(4xz \cdot 2z = 8xz^{1+ 1}=8xz^2\) (using the rule of exponents \(a^m \cdot a^n = a^{m + n}\) for \(z\)).
So we get \(\frac{12x^3z^2}{8xz^2}\).

Step2: Simplify the fraction

Now we simplify \(\frac{12x^3z^2}{8xz^2}\) by canceling out the common factors in the numerator and the denominator.

• For the coefficients: \(\frac{12}{8}=\frac{3}{2}\) (dividing numerator and denominator by their greatest common divisor, which is 4).
• For the \(x\) terms: \(\frac{x^3}{x}=x^{3 - 1}=x^2\) (using the rule of exponents \(\frac{a^m}{a^n}=a^{m - n}\)).
• For the \(z\) terms: \(\frac{z^2}{z^2}=1\) (since any non - zero number to the power of 0 is 1, and \(z^2\div z^2 = z^{2-2}=z^0 = 1\)).

Multiplying these simplified parts together, we have \(\frac{3}{2}\cdot x^2\cdot1=\frac{3x^2z}{2}\).

Answer:

\(\frac{3x^2z}{2}\) (corresponding to the option \(\boldsymbol{\frac{3x^2z}{2}}\))