simpity the iollowing \( \frac{x^{12}}{x^{7}} \) \( \frac{x^{7}}{x^{12}} \) \( \frac{4^{6}}{4^{9}} \) (e) \( \frac{0^{6} b^{8} c}{a^{8} b^{6} c} \) Simplify the following (d) \( \left(x^{3}\right)^{4} \) (b) \( \left(x^{2}\right)^{5} \) (a) \( (x)^{3} \) (e) \( \left(3^{2}\right)^{3} \) (g) \( \quad\left(4 x^{3}\right)^{3} \) (li) \( \left(7 a^{7}\right)^{2} \) (j) \( \left(-4 x^{3}\right)^{4} \) (k) \( \left(x^{4}\right)^{2} \) (px) \( \left(x^{4}\right)^{2} \cdot\left(x^{2}\right)^{4} \) (8) \( \quad\left(3 x^{4}\right)^{2}+\left(2 x^{2}\right)^{4} \) (in) (d) \( \left(3 x^{4}\right)^{2} \cdot\left(2 x^{2}\right)^{4} \) (p) Simplify the following. (r) (b)

1. Simplify \(\frac{x^{12}}{x^{7}}\):   \[   \frac{x^{12}}{x^{7}}=x^{12-7}=x^5.   \] 2. Simplify \(\frac{x^{7}}{x^{12}}\):   \[   \frac{x^{7}}{x^{12}}=x^{7-12}=x^{-5}=\frac{1}{x^5}.   \] 3. Simplify \(\frac{4^{6}}{4^{9}}\):   \[   \frac{4^{6}}{4^{9}}=4^{6-9}=4^{-3}=\frac{1}{4^3}=\frac{1}{64}.   \] 4. Simplify \(\frac{0^{6}\,b^{8}\,c}{a^{8}\,b^{6}\,c}\):   Since \(0^6=0\) (assuming \(a\ne0\), \(b\ne0\), and \(c\ne0\)), the numerator is 0. Thus,   \[   \frac{0^{6}\,b^{8}\,c}{a^{8}\,b^{6}\,c}=0.   \] 5. Simplify \(\left(x^{3}\right)^{4}\):   \[   \left(x^{3}\right)^{4}=x^{3\cdot 4}=x^{12}.   \] 6. Simplify \(\left(x^{2}\right)^{5}\):   \[   \left(x^{2}\right)^{5}=x^{2\cdot 5}=x^{10}.   \] 7. Simplify \((x)^{3}\):   \[   (x)^{3}=x^3.   \] 8. Simplify \(\left(3^{2}\right)^{3}\):   \[   \left(3^{2}\right)^{3}=3^{2\cdot 3}=3^6.   \]   (Optional numerical evaluation: \(3^6=729\).) 9. Simplify \(\left(4x^{3}\right)^{3}\):   Use the rule \((ab)^n=a^n\,b^n\):   \[   \left(4x^{3}\right)^{3}=4^3\,(x^3)^3=64\,x^{9}.   \] 10. Simplify \(\left(7a^{7}\right)^{2}\):   \[   \left(7a^{7}\right)^{2}=7^2\,(a^7)^2=49\,a^{14}.   \] 11. Simplify \(\left(-4x^{3}\right)^{4}\):   Since the exponent is even, the negative sign vanishes:   \[   \left(-4x^{3}\right)^{4}=(-4)^4\,(x^3)^4=256\,x^{12}.   \] 12. Simplify \(\left(x^{4}\right)^{2}\):   \[   \left(x^{4}\right)^{2}=x^{4\cdot 2}=x^8.   \] 13. Simplify \(\left(x^{4}\right)^{2}\cdot\left(x^{2}\right)^{4}\):   First, simplify each factor:   \[   \left(x^{4}\right)^{2}=x^8,\quad \left(x^{2}\right)^{4}=x^{8}.   \]   Then,   \[   x^{8}\cdot x^{8}=x^{8+8}=x^{16}.   \] 14. Simplify \(\left(3x^{4}\right)^{2}+\left(2x^{2}\right)^{4}\):   Simplify each term separately:   \[   \left(3x^{4}\right)^{2}=3^2\,x^{8}=9x^8,   \]   \[   \left(2x^{2}\right)^{4}=2^4\,x^{8}=16x^8.   \]   Then,   \[   9x^8+16x^8=25x^8.   \] 15. Simplify \(\left(3x^{4}\right)^{2}\cdot\left(2x^{2}\right)^{4}\):   Simplify each factor:   \[   \left(3x^{4}\right)^{2}=9x^8,\quad \left(2x^{2}\right)^{4}=16x^8.   \]   Multiply them:   \[   9x^8 \cdot 16x^8=144\,x^{16}.   \] Any parts labeled \((p)\), \((r)\), or \((b)\) that do not have a corresponding expression were not provided.