The rational expressions [(x² + 16xz + 64z²)/(x²z + 8xz²)] ÷ [(x² + 5xz + 24z²)/(x⁴z - 9x²z³)] can be simplified as; x(x - 3z)
<h3>How to divide algebraic expressions?</h3>
We want to divide the rational expressions;
[(x² + 16xz + 64z²)/(x²z + 8xz²)] ÷ [(x² + 5xz + 24z²)/(x⁴z - 9x²z³)]
Let us break down the expressions one after the other;
(x² + 16xz + 64z²) can be expressed as;
(x + 8z)²
Likewise, (x²z + 8xz²) can be expressed as;
xz(x + 8z)
Thus;
[(x² + 16xz + 64z²)/(x²z + 8xz²)] is;
(x + 8z)²/(xz(x + 8z)) = (x + 8z)/xz
(x² + 5xz + 24z²) can be expressed as;
(x + 3z)(x + 8z)
Similarly;
(x⁴z - 9x²z³) can be expressed as;
x²z(x² - 9z²)
Thus;
[(x² + 5xz + 24z²)/(x⁴z - 9x²z³)] can be expressed as;
(x + 3z)(x + 8z)/(x²z(x² - 9z²))
Then our main expression can now be expressed as;
[(x + 8z)/xz] ÷ (x + 3z)(x + 8z)/(x²z(x² - 9z²))
This can be rewritten as;
[(x + 8z)/xz] * [(x²z(x² - 9z²))]/((x + 3z)(x + 8z))
x(x² - 9z²)/(x + 3z)
This can further be simplified as;
x(x + 3z)(x - 3z)/(x + 3z) = x(x - 3z)
Read more about Algebraic expressions at; brainly.com/question/723406
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