I4.1 Let $\alpha$ be the remainder of $2^{1000}$ divided by 13 , determine the value of $\alpha$.
I4.2 Determine the value of $\beta=\frac{(7+4 \sqrt{\alpha})^{\frac{1}{2}}-(7-4 \sqrt{\alpha})^{\frac{1}{2}}}{\sqrt{\alpha}}$.
I4.3 If $f(a)=a-\beta$ and $F(a, b)=b^{2}+a$, determine the value of $\gamma=F(3, f(4))$.
I4.4 If $\delta$ is the product of all real roots of $x^{\log _{7} x}=10$, determine the value of $\delta$.
$\begin{array}{l}13 \times 5=64+1 \Rightarrow 2^{6}=13 \times 5-1 \\ 2^{1000}=2^{4} \cdot 2^{996}=16 \cdot\left(2^{6}\right)^{166}=(13+3) \cdot(13 \times 5-1)^{166} \\ =(13+3) \cdot(13 m+1) \text {, by using binomial theorem } \\ =13 n+3 \text {, where } n \text { and } m \text { are integers } \\ \alpha=3 \\ \sqrt{7+4 \sqrt{3}}=\sqrt{7+2 \sqrt{12}}=\sqrt{4+3+2 \sqrt{4 \times 3}}=\sqrt{4}+\sqrt{3}=2+\sqrt{3} \\ \sqrt{7-4 \sqrt{3}}=2-\sqrt{3} \\ \beta=\frac{(7+4 \sqrt{3})^{\frac{1}{2}}-(7-4 \sqrt{3})^{\frac{1}{2}}}{\sqrt{3}}=\frac{2+\sqrt{3}-2+\sqrt{3}}{\sqrt{3}}=2 \\ f(4)=4-2=2 \\ \gamma=F(3, f(4))=F(3,2)=2^{2}+3=7 \\ x^{\log _{7} x}=10 \\ \log _{7} x \log x=\log 10 \\ \frac{(\log x)^{2}}{\log 7}=1 \\ \log x= \pm \sqrt{\log 7} \\ x=10^{\sqrt{\log 7}} \text { or } 10^{-\sqrt{\log 7}} \\ \text { Product of roots }=10^{\sqrt{\log 7}} \times 10^{-\sqrt{\log 7}}=1 \\\end{array}$