Simple question on factoring the difference of 2 perfect squares

Question

(b) (i) Use the identity $A^2-B^2=(A-B)(A+B)$ to factorise the expression $5^{2k}-1$.

Do I just put the k as 1 so that the equation is 5^2 and 1^2

Thanks

Steve

Answer 1

For all real numbers A and B, $$A^2-B^2=(A-B)(A+B)$$ Substituting $$A=5^k$$ and $$B=1$$, $$(5^k)^2-1^2=(5^k-1)(5^k+1) \implies 5^{2k}-1= (5^k-1)(5^k+1)$$ Note that 5^k-1 can be further factorised if k is a natural number. In that case, $$5^k-1=(5-1)(1+5^1+5^2+5^3+......+5^{k-1}) = 4*(1+5^1+5^2+5^3+......+5^{k-1})$$