Find P 511996

1. **State the problem:** We need to find the value of $p$ such that in the expansion of $$\left(3x + \frac{p}{x}\right)^8$$ the coefficients of the terms with $x^4$ and $x^6$ are equal. 2. **Recall the binomial expansion formula:** $$\left(a + b\right)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$$ 3. **Identify terms:** Here, $a = 3x$ and $b = \frac{p}{x}$, and $n=8$. 4. **General term:** The $k$-th term (starting from $k=0$) is $$T_{k+1} = \binom{8}{k} (3x)^{8-k} \left(\frac{p}{x}\right)^k = \binom{8}{k} 3^{8-k} p^k x^{8-k-k} = \binom{8}{k} 3^{8-k} p^k x^{8-2k}$$ 5. **Find $k$ for $x^4$ term:** We want the power of $x$ to be 4: $$8 - 2k = 4 \implies 2k = 4 \implies k = 2$$ 6. **Find $k$ for $x^6$ term:** Similarly, for power 6: $$8 - 2k = 6 \implies 2k = 2 \implies k = 1$$ 7. **Write coefficients for these terms:** - Coefficient of $x^4$ term (when $k=2$): $$C_4 = \binom{8}{2} 3^{6} p^{2}$$ - Coefficient of $x^6$ term (when $k=1$): $$C_6 = \binom{8}{1} 3^{7} p^{1}$$ 8. **Set coefficients equal:** $$\binom{8}{2} 3^{6} p^{2} = \binom{8}{1} 3^{7} p$$ 9. **Substitute binomial coefficients:** $$28 \cdot 3^{6} p^{2} = 8 \cdot 3^{7} p$$ 10. **Divide both sides by $p$ (since $p>0$):** $$28 \cdot 3^{6} p = 8 \cdot 3^{7}$$ 11. **Divide both sides by $3^{6}$:** $$28 p = 8 \cdot 3$$ 12. **Simplify:** $$28 p = 24$$ 13. **Solve for $p$:** $$p = \frac{24}{28} = \frac{6}{7}$$ **Final answer:** $$p = \frac{6}{7}$$