Question

Simplify each expression. $$\sqrt{90(x+4)^{3}}$$

   Simplify each expression.
$$\sqrt{90(x+4)^{3}}$$

Introductory Algebra for College Students

Introductory Algebra for College Students

Robert Blitzer 6th Edition

Chapter 8, Problem 109

Instant Answer

Solved by Expert Narayan Hari

02/11/2021

Step 1

The product rule states that $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$. So, we can rewrite $\sqrt{90(x+4)^{3}}$ as $\sqrt{9 \cdot 10(x+4)^{2} \cdot (x+4)}$. Show more…

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Simplify each expression. $$\sqrt{90(x+4)^{3}}$$

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Key Concepts

Exponent Rules in Radical Expressions

When expressions involving exponents appear under a radical, it is useful to express exponents as products of exponents that match the index of the radical (for square roots, groups of two). This allows rewriting expressions (such as rewriting an exponent into a squared term and another exponent) to simplify the radical, maintaining the rules of exponents while accounting for absolute values when necessary.

Properties of Radicals

This concept covers the rules that allow the simplification of radical expressions. It includes the product rule for radicals (i.e., ?(ab) = ?a ?b for nonnegative a and b) and the understanding that the square root of a square gives the absolute value of the original expression. These properties are fundamental when breaking down and simplifying radical expressions.

Extracting Perfect Squares

Extracting perfect square factors from under a radical is key to simplifying an expression. This involves identifying factors within the radicand that are perfect squares, so they can be taken out of the radical. Recognizing and separating these factors reduces the complexity of the expression.

Factorization of Integers

Factorization involves breaking down composite numbers into their prime components. In the context of radical simplification, factoring allows one to group elements into perfect squares, which can then be extracted from the square root. This method is essential for reducing numerical radicands.

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