Can anyone help me to prove these identities?
\(\displaystyle \mbox{(a) }\, \big|\,x^2\,\big|\, =\, \big|\,x\,\big|^2\)
\(\displaystyle \mbox{(b) }\, \big|\, x^n\, \big|\, =\, \big|\, x\, \big|^n\, \mbox{ for every integer }\, n\)
\(\displaystyle \mbox{(c) }\, \big|\, x\, \big|\, =\, \sqrt{\strut x^2\,}\)
\(\displaystyle \mbox{(d) }\, \big|\, x\, -\, y\,\big|\, \leq\, \big|\, x\, \big|\, +\, \big|\, y\, \big|\)
\(\displaystyle \mbox{(e) }\, \big|\, x\, -\, y\, \big|\, \geq\, \bigg|\big|\, x\, \big|\, -\, \big|\, y\, \big|\bigg|\)
Hint: In (e), prove that \(\displaystyle \big|\, x\, -\, y\, \big|\, \geq\, \big|\, x\, \big|\, -\, \big|\, y\, \big|\) and \(\displaystyle \big|\, x\, -\, y\, \big|\, \geq\, \big|\, y\, \big|\, -\, \big|\, x\, \big|\)
\(\displaystyle \mbox{(a) }\, \big|\,x^2\,\big|\, =\, \big|\,x\,\big|^2\)
\(\displaystyle \mbox{(b) }\, \big|\, x^n\, \big|\, =\, \big|\, x\, \big|^n\, \mbox{ for every integer }\, n\)
\(\displaystyle \mbox{(c) }\, \big|\, x\, \big|\, =\, \sqrt{\strut x^2\,}\)
\(\displaystyle \mbox{(d) }\, \big|\, x\, -\, y\,\big|\, \leq\, \big|\, x\, \big|\, +\, \big|\, y\, \big|\)
\(\displaystyle \mbox{(e) }\, \big|\, x\, -\, y\, \big|\, \geq\, \bigg|\big|\, x\, \big|\, -\, \big|\, y\, \big|\bigg|\)
Hint: In (e), prove that \(\displaystyle \big|\, x\, -\, y\, \big|\, \geq\, \big|\, x\, \big|\, -\, \big|\, y\, \big|\) and \(\displaystyle \big|\, x\, -\, y\, \big|\, \geq\, \big|\, y\, \big|\, -\, \big|\, x\, \big|\)
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