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. $- 3 \sqrt{5 p^{3}}$

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Identify the given expression: $-3 \sqrt{5p^{3}}$. Notice that it is in radical form because of the square root symbol $\sqrt{\cdot}$.

Recall that the square root of a number or expression can be written as an exponent of $\frac{1}{2}$. So, $\sqrt{a} = a^{\frac{1}{2}}$.

Rewrite the radical expression $\sqrt{5p^{3}}$ in exponential form as $(5p^{3})^{\frac{1}{2}}$.

Apply the exponent to each factor inside the parentheses separately using the property $(ab)^{m} = a^{m} b^{m}$, so $(5p^{3})^{\frac{1}{2}} = 5^{\frac{1}{2}} \cdot (p^{3})^{\frac{1}{2}}$.

Simplify the powers of $p$ by multiplying the exponents: $(p^{3})^{\frac{1}{2}} = p^{3 \times \frac{1}{2}} = p^{\frac{3}{2}}$. The expression in exponential form is then $-3 \cdot 5^{\frac{1}{2}} \cdot p^{\frac{3}{2}}$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Exponential and Radical Forms

Exponential form expresses roots using fractional exponents, where the nth root of a number is written as that number raised to the power 1/n. Radical form uses the root symbol (√) to denote roots. Converting between these forms involves rewriting roots as fractional exponents or vice versa.

Properties of Exponents

Understanding how to manipulate exponents is essential, including the rules for multiplying powers, raising powers to powers, and handling fractional exponents. For example, (a^m)^n = a^(m*n) and a^(1/n) represents the nth root of a. These properties help simplify expressions during conversion.

Evaluating Expressions with Positive Variables

Since variables represent positive real numbers, expressions involving roots and exponents are well-defined and real-valued. This assumption allows simplification without considering complex numbers or absolute values, ensuring the evaluation of expressions like roots and powers is straightforward.

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