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If I have Question like that :

If $ x=\frac{-1}{2} \ , \ y=\frac{3}{4} \ , \ z=\frac{-3}{2} \ $

Find a numerical Value for $$x^3 \div y^2z^2$$

First If we divide $x^3$ over $y^2$ and then multiplying the result we get $\frac{-1}{2}$ , but if I think we should first Multiply $y^2z^2$ and then we divide $x^3$ over the result , and the answer in that case is $\frac{-8}{81}$

First We notice that $$z=-2y \ \ , x=\frac{-2}{3}y$$ So $$x^3 \div y^2z^2 = (\frac{-2}{3})^3 \times y^3 \div 4\ y^4 = \frac{-8}{81} $$

For example $$6 \div 9 = \frac{2}{3}$$ And Notice if we wrote $$6 \div 9 = 6 \div 3 \times 3 = 6 \div 3^2$$ Notice $6 \div 9 = \frac{2}{3}$ but $6 \div 3 \times 3 = 6$

The write way is $$6 \div 9 =6 \div (3 \times 3) , 6 \div 3 \times 3 \neq 6 \div 9$$ And $$6\div 3^2 \neq 6 \div 3\times 3 \ \text{but} \ 6 \div 3^2 = 6 \div (3 \times 3)$$

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Find a numerical Value for $$x^3 \div y^2z^2$$

I find this notation a bit ambiguous, although I would assume the intended interpretation is: $$\frac{x^3}{y^2z^2} \tag{$*$}$$ The division operator $\div$ isn't used much in mathematics (at least not 'later on') and if you would write something like $x^3/y^2z^2$, the strict interpretation would be: $$\frac{x^3}{y^2}z^2 \tag{$\star$}$$ because division and multiplication are at the same level and you apply them as they appear from left to right. Almost all (mathematical) software will interpret $x^3/y^2z^2$ in this way, although when written by humans, some will mean $(*)$ but this is to be avoided.

When you can use fractions, such as in $(*)$ and $(\star)$, no confusion is possible. If you're forced to write 'in line', then parentheses can be used to make sure $x^3/(y^2z^2)$ gives $(*)$, at least if that's what you mean. If you actually mean $(\star)$, then although $x^3/y^2z^2$ is technically correct, writing it as $x^3z^2/y^2$ or as $x^3z^2 \div y^2$ is a lot clearer and avoids this ambiguity and possible confusion.

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