Simplify each radical expression. Assume that all variables represent positive numbers. √(72 a^3

step 1 Now we have to simplify square root of 72 a cube b power 5 and c Now we can write 72 as 36 times

Simplify each radical expression. Assume that all variables...

Key Concepts

Radical Expression

A radical expression involves the use of a radical sign to denote the extraction of a root, such as a square root or cube root. It represents a number or expression that is not necessarily written in its simplest form until simplification methods are applied.

Simplifying Radicals

Simplifying radicals means rewriting a radical expression in its simplest form by extracting perfect power factors from under the radical sign. This process ensures that no perfect square (or other appropriate power for different types of roots) remains under the radical, resulting in a more compact and standardized representation.

Product Property of Square Roots

The product property of square roots states that the square root of a product is equal to the product of the square roots of the individual factors, provided that all variables and expressions involved represent nonnegative numbers. This property is crucial for breaking down a complex radical into manageable parts that can be simplified separately.

Prime Factorization and Identification of Perfect Powers

Using prime factorization involves breaking down a numerical coefficient into its prime factors to identify perfect squares or other perfect powers. Recognizing these perfect powers allows for their extraction from under the radical, thereby simplifying the overall expression.

Laws of Exponents in the Context of Radicals

When simplifying radical expressions involving variables, the laws of exponents are used to separate the exponent into parts that form a perfect power and a remainder. This enables the extraction of the variable part that is a perfect power outside the radical, which is an essential step in the simplification process.

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