To solve the expression [tex]\(\left|-24 x^7 y^6 z^3\right| \div\left|12 x^3 y^3 z\right|\)[/tex], we follow a step-by-step approach:
1. Simplify the constants:
- First, we take the absolute values of the constants: [tex]\(|-24|\)[/tex] and [tex]\(|12|\)[/tex].
- The absolute value of -24 is 24, and the absolute value of 12 is 12.
- Next, we divide these constants: [tex]\( \frac{24}{12} = 2.0 \)[/tex].
2. Simplify the variable [tex]\(x\)[/tex]:
- We need to divide the terms involving [tex]\(x\)[/tex]: [tex]\(x^7\)[/tex] by [tex]\(x^3\)[/tex].
- When dividing exponential terms with the same base, subtract the exponents: [tex]\(x^{7-3} = x^4\)[/tex].
3. Simplify the variable [tex]\(y\)[/tex]:
- We need to divide the terms involving [tex]\(y\)[/tex]: [tex]\(y^6\)[/tex] by [tex]\(y^3\)[/tex].
- Similarly, by subtracting the exponents: [tex]\(y^{6-3} = y^3\)[/tex].
4. Simplify the variable [tex]\(z\)[/tex]:
- We need to divide the terms involving [tex]\(z\)[/tex]: [tex]\(z^3\)[/tex] by [tex]\(z\)[/tex].
- Again, by subtracting the exponents: [tex]\(z^{3-1} = z^2\)[/tex].
Putting it all together, we get:
[tex]\[
\left|-24 x^7 y^6 z^3\right| \div\left|12 x^3 y^3 z\right| = 2.0 x^4 y^3 z^2
\][/tex]
Thus, the simplified form of the given expression is [tex]\(\boxed{2.0 x^4 y^3 z^2}\)[/tex].
