BigGlenntheHeavy said:Without using a calculator, just pencil and paper, find the simplified numerical value of:
\(\displaystyle \sqrt{7+\sqrt{48}}+\sqrt{7-\sqrt{48}} \ = \ ?\)
Hint; Answer is an integer.
BigGlenntheHeavy said:\(\displaystyle Good \ show \ Subhotosh \ Khan, \ as \ I \ didn't \ go \ that \ way, \ here's \ what \ I \ did.\)
\(\displaystyle \sqrt{7+\sqrt{48}} \ = \ [7+4\sqrt3]^{1/2} \ = \ [4+4\sqrt3+3]^{1/2} \ = \ [2^2+4\sqrt3+(\sqrt3)^2]^{1/2}\)
\(\displaystyle =[(2+\sqrt3)^2]^{1/2} \ = \ 2+\sqrt3, \ \sqrt{7-\sqrt{48}} \ = \ 2-\sqrt3\)
\(\displaystyle Hence, \ \sqrt{7+\sqrt{48}} \ + \ \sqrt{7-\sqrt{48}} \ = \ 2+\sqrt3+2-\sqrt3 \ = \ 4\)
\(\displaystyle Only \ problem \ with \ your \ method \ is \ that \ you \ pick \ up \ an \ extraneous \ root.\)
BigGlenntheHeavy said:\(\displaystyle Sorry, \ but \ I \ would \ not \ as \ (2+\sqrt3) \ is \ a \ constant.\)
\(\displaystyle \sqrt(x^2) \ = \ |x|, \ but \ \sqrt(5^2) \ = \ 5.\)
\(\displaystyle The \ reason \ the \ \sqrt(x^2) \ = \ |x| \ is \ we \ don't \ know \ whether \ x \ is \ positive \ or \ negative.\)
\(\displaystyle but \ we \ always \ know \ the \ value \ of \ a \ constant.\)
\(\displaystyle After thought: \ [(2+\sqrt3)^2]^{1/2} \ = \ [4+4\sqrt3+3]^{1/2} \ = \ [7+4\sqrt3]^{1/2}\)
\(\displaystyle Now, \ how \ is \ [7+4\sqrt3]^{1/2} \ negative?\)
\(\displaystyle In \ fact, \ to \ be \ mathematically \ correct, \ your \ answer \ is \ wrong.\)
BigGlenntheHeavy said:\(\displaystyle Your \ answer \ should \ \ have \ been:\)
\(\displaystyle y^2 \ = \ 16, \ |y| \ =4, \ y \ = \ 4, \ y \ \ge0 \ (the \ sum \ of \ two \ square \ roots).\)