Explain why the equation [ sqrt x^8 = x^4 ] is valid for all real numbers x, with no necessity for using absolute value.
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Both x^8 and x^4 have even exponents so they're never negative. The square root is never negative (unless you write ±√x^8). So x can be any real number, and neither side of this equation will ever be negative. Therefore, absolute value is not needed.