If the expression is in exponential form, write it in radical form and evaluate if possible. If it is in radical form, write it in exponential form. Assume all variables represent positive real numbers. -12x^1/2

Verified step by step guidance

1. Identify the form of the given expression. The expression is \(-12x^{1/2}\), which is in exponential form because the variable \(x\) is raised to a fractional exponent \(\frac{1}{2}\).
2. Recall the relationship between exponents and radicals: \(x^{m/n} = \sqrt[n]{x^m}\). Specifically, \(x^{1/2} = \sqrt{x}\).
3. Rewrite the expression \(-12x^{1/2}\) in radical form by replacing \(x^{1/2}\) with \(\sqrt{x}\). This gives \(-12\sqrt{x}\).
4. Since the expression is now in radical form, check if it can be simplified or evaluated further. Because \(x\) is a variable representing a positive real number and no specific value is given, the expression \(-12\sqrt{x}\) is the simplified radical form.
5. If you were to convert back to exponential form, you would write \(-12\sqrt{x}\) as \(-12x^{1/2}\), confirming the equivalence of the two forms.

Key Concepts

Exponential and Radical Forms

Exponential form expresses roots using fractional exponents, such as x^(1/2) for the square root of x. Radical form uses root symbols, like √x for the square root. Converting between these forms involves rewriting fractional exponents as radicals and vice versa.

Properties of Exponents

Understanding how to manipulate exponents is essential, especially fractional exponents which represent roots. For example, x^(m/n) equals the nth root of x raised to the mth power. This helps in rewriting expressions and simplifying them correctly.

Evaluating Expressions with Positive Variables

Since variables represent positive real numbers, roots and fractional powers are defined and real. This assumption allows safe evaluation of expressions like x^(1/2) without considering complex numbers or negative roots.