Multiple Choice
Determine if the given square root evaluates to a real number.
11. Roots, Radicals, and Complex Numbers
Introduction to Radical Expressions
Multiple Choice
Simplify the root.
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Imaginary
Verified step by step guidance1
Recognize that the expression involves the fourth root of a negative number: \(^4\sqrt{-625}\). Since 625 is positive, the negative sign indicates the number is negative inside the root.
Recall that even roots (like the fourth root) of negative numbers are not real numbers, but imaginary or complex numbers. This is because any real number raised to an even power is non-negative.
Rewrite the expression by separating the negative sign using the imaginary unit \(i\), where \(i = \sqrt{-1}\). So, \(^4\sqrt{-625} = ^4\sqrt{625 \times (-1)} = ^4\sqrt{625} \times ^4\sqrt{-1}\).
Calculate \(^4\sqrt{625}\) by expressing 625 as a power of a number: \$625 = 5^4$. Therefore, \(^4\sqrt{625} = 5\).
Understand that \(^4\sqrt{-1}\) is the fourth root of \(-1\), which is a complex number. Since the fourth root of \(-1\) is not a real number, the entire expression is imaginary.
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