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Chapter 10: Problem 22

Assume for all exercises that even roots are of non- negative quantities and that all denominators are nonzero. Write an equivalent expression using radical notation and, if possible, simplify. $$ b^{3 / 2} $$

Short Answer

Expert verified
The equivalent expression is \( b \sqrt{b} \).

Step by step solution

01

Understand the Exponent

The given expression is a power expression: \(b^{3/2}\). The exponent \(3/2\) is a fractional exponent.
02

Convert to Radical Notation

A fractional exponent \(a/b\) can be written as a radical expression. Specifically, \(x^{a/b} = \sqrt[b]{x^a}\). So \(b^{3/2}\) can be written in radical notation as \(\sqrt{b^3}\).
03

Simplify the Expression

Simplify the radical expression if possible. In this case, \(\sqrt{b^3} = \sqrt{b^2 \cdot b} = b \sqrt{b}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

radical notation
Radical notation is a way to express roots using the radical symbol \(\sqrt{}\). For instance, the square root of \(x\) is written as \(\sqrt{x}\). When dealing with fractions in exponents, it is useful to convert these into radical notation. For example, the expression \(b^{3/2}\) can be rewritten in radical form. Here's how we do it:
simplify radical expressions
To simplify a radical expression, we need to break it down into its simplest components. Let's simplify \(\sqrt{b^3}\):
  • First, rewrite \(b^3\) as \(b^2 \cdot b\).
  • Next, use the property \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\).
  • This gives us \(\sqrt{b^2 \cdot b} = \sqrt{b^2} \cdot \sqrt{b} = b \cdot \sqrt{b}\).

The simplified expression in radical form is \(b \sqrt{b}\).
exponent rules
Understanding exponent rules is crucial for mathematical proficiency. Let's look at some key rules.
  • Product Rule: \(a^m \cdot a^n = a^{m+n}\).
  • Power Rule: \((a^m)^n = a^{m\cdot n}\).
  • Fractional Exponents: \(a^{1/n} = \sqrt[n]{a}\) and \(a^{m/n} = \sqrt[n]{a^m}\).

For our problem, we started with \(b^{3/2}\). Using the fractional exponent rule, we know \(b^{3/2} = \sqrt{b^3}\). Applying the power rule further lets us simplify to \(b \sqrt{b}\).

These combined rules make dealing with complex expressions more approachable.

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Most popular questions from this chapter

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