Question

7.3 ha2
guided solution
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1. find the quotient. if possible, write your answer in factored form.

\\(\frac{2xyz}{x^3z^3} div \frac{6y^4}{2x^2z^2} = \boxed{\frac{2}{3y^3}}\\) , \\(x
eq square\\) , \\(z
eq square\\)

Explanation:

Step1: Analyze the denominators

In the given rational expression \(\frac{2xyz}{x^{3}z^{3}}\div\frac{6y^{4}}{2x^{2}z^{2}}\), we need to find the values of \(x\) and \(z\) for which the expression is undefined. A rational expression is undefined when its denominator is zero.
For the first fraction \(\frac{2xyz}{x^{3}z^{3}}\), the denominator is \(x^{3}z^{3}\). Setting \(x^{3}z^{3}=0\), we know that if either \(x = 0\) or \(z = 0\), the denominator becomes zero.
For the second fraction \(\frac{6y^{4}}{2x^{2}z^{2}}\), the denominator is \(2x^{2}z^{2}\). Setting \(2x^{2}z^{2}=0\), again if \(x = 0\) or \(z = 0\), the denominator is zero. Also, when we perform division of fractions, we multiply by the reciprocal, so we have \(\frac{2xyz}{x^{3}z^{3}}\times\frac{2x^{2}z^{2}}{6y^{4}}\). The denominators from both fractions (before and after reciprocal) contribute to the undefined points.
So, to find the restrictions, we look at the values that make any of the denominators zero.
For \(x\): In \(x^{3}z^{3}\) and \(2x^{2}z^{2}\), when \(x = 0\), the denominators are zero.
For \(z\): In \(x^{3}z^{3}\) and \(2x^{2}z^{2}\), when \(z = 0\), the denominators are zero.

Step2: Determine the restrictions

So, the expression is undefined when \(x = 0\) or \(z = 0\). Therefore, \(x
eq0\) and \(z
eq0\).

Answer:

For \(x
eq\boldsymbol{0}\) and \(z
eq\boldsymbol{0}\)