Algebra
Advanced Factorization
Evaluate \[\frac{(167 ^2-165 ^2)(163 ^2-161 ^2)(159 ^2-157 ^2)}{166 \times 162 \times 158}.\]
Evaluate \[\sqrt{23 \times 24 \times 25 \times 26+1}.\]
\(\quad\) What is the real number \(A\) that satisfies
\[\frac{77 ^4+77 ^2+1}{77 ^2+77+1}=78 ^2-A?\]
What is the value of \[490000-\frac{490000 \times 490001+1}{490701}?\]
If \(a\), \(c\) and \(e\) are prime numbers such that \(a<c<e\) and \[3^{6}+3^{7}+3^{8}+3^{9}=a^b c^d e^f,\] what is the value of \(bdf\)?