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 posi-tive real numbers. -3 √5p³

Verified step by step guidance

1. Identify the given expression: \(-3 \sqrt{5p^3}\).
2. Recognize that the expression is in radical form, specifically a square root.
3. Convert the square root into exponential form: \(\sqrt{a} = a^{1/2}\).
4. Apply the conversion to the expression: \(-3 \sqrt{5p^3} = -3 (5p^3)^{1/2}\).
5. Distribute the exponent to both the constant and the variable: \(-3 \cdot 5^{1/2} \cdot (p^3)^{1/2}\).

Key Concepts

Exponential Form

Exponential form expresses numbers using a base raised to a power, such as a^b, where 'a' is the base and 'b' is the exponent. This notation is useful for simplifying calculations, especially with roots and powers. For example, the expression 5^(1/2) represents the square root of 5, indicating that the exponent is a fraction.

Radical Form

Radical form represents numbers using roots, such as √a or n√a, where 'n' indicates the degree of the root. This form is particularly useful for expressing non-integer exponents and can be converted to exponential form. For instance, the expression √5 can be rewritten as 5^(1/2), demonstrating the relationship between radicals and exponents.

Positive Real Numbers

Positive real numbers are all the numbers greater than zero, including integers, fractions, and irrational numbers. In algebra, assuming variables represent positive real numbers ensures that operations involving roots and exponents yield real results, avoiding issues with undefined expressions. This assumption is crucial when evaluating expressions like -3√5p³, as it affects the validity of the results.