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Chapter 7: Problem 14

Write each exponential expression using radical notation. \((-b)^{1 / 5}\)

Short Answer

Expert verified
\root{5}{-b}

Step by step solution

01

Understand the Exponential Expression

The given expression is \((-b)^{1 / 5}\). This is an exponential expression with a fractional exponent.
02

Apply the Radical Notation Rule

A fractional exponent \(\frac{1}{n}\) can be converted into radical notation as the \(\text{n-th}\) root. For example, \((x)^{1 / n}\) can be written as \(\root{n}{x}\).
03

Convert the Fractional Exponent to Radical Notation

Use the rule from Step 2 to convert the expression \((-b)^{1 / 5}\). Writing it in radical notation gives us \(\root{5}{-b}\).

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Key Concepts

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

Fractional Exponent
When you see a fractional exponent, it might look a bit confusing at first. Don't worry, it's simpler than it appears. A fractional exponent is an exponent that is a fraction instead of a whole number. For example, in the expression \((-b)^{1 / 5}\), the exponent \(\frac{1}{5}\) is a fraction.
The numerator (top part) of the fraction is 1, and the denominator (bottom part) is 5. To help you understand better, remember these points:
  • A fractional exponent represents both a power and a root simultaneously.
  • The numerator indicates the power, which is 1 in this case.
  • The denominator tells you the root, which is 5 here.
So, using a fractional exponent makes it easy to handle both root and power operations together in one step.
Exponential Expression
An exponential expression involves numbers raised to a power, or exponent. Let's break down \((-b)^{1 / 5}\) to understand it better. Here are a couple of key points:
  • The base of the expression is \(-b\).
  • The exponent, \(\frac{1}{5}\), tells you to apply the fifth root to the base.
An exponent tells you how many times to multiply the base by itself. When it's a fraction, it's like combining multiplication and division (or roots). In this expression, using a fractional exponent morphs \(-b\) under the influence of a root (fifth root in this case). So instead of multiplying \(-b\) by itself, you are finding a value that, when multiplied by itself 5 times, equals \-b\.
Root Conversion
Converting a fractional exponent to radical notation means changing the fraction into a root. For instance, to convert \((-b)^{1 / 5}\) to radical form, remember these steps:
First, identify the denominator of the fraction (which is 5 here). This tells you the type of root you need (a fifth root). So, \((-b)^{1 / 5}\) becomes \(\root{5}{-b}\).
Here's a simple way to visualize it:
  • Fractional Exponent: \((-b)^{1 / 5}\)
  • Find the root as per the denominator: Fifth root
  • Radical Notation: \(\root{5}{-b}\)
To recap, by converting fractional exponents to radical notation, we make it easier to understand and solve expressions involving roots and exponents more intuitively.

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Most popular questions from this chapter

Solve each equation. $$\sqrt{5 x^{2}+4 x+1}-x=0$$

Simplify. $$\sqrt[3]{m}(\sqrt[3]{m^{2}}-\sqrt[3]{m^{5}})$$

Simplify. $$\sqrt{3 a b}(\sqrt{3 a}+\sqrt{3})$$

Use a calculator to find approximate solutions to the following equations. Round your answers to three decimal places. $$x^{2 / 3}=8.86$$

Simplify. $$\frac{\sqrt{6}}{2} \cdot \frac{1}{\sqrt{3}}$$
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