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Chapter 11: Problem 38

Write each expression in simplest radical form. If \(a\) radical appears in the denominator, rationalize the denominator. $$\sqrt[9]{-27}$$

Short Answer

Expert verified
\( \sqrt[9]{-27} = \sqrt[3]{-3} \)

Step by step solution

01

Simplify the radical

The expression to simplify is \( \sqrt[9]{-27} \). Notice that -27 can be written as \( (-3)^3 \). We want to express this under a different root if possible.
02

Rewrite under the ninth root

We have \( \sqrt[9]{(-3)^3} \). According to the properties of exponents, \( \sqrt[9]{a^b} = a^{b/9} \). Therefore, \( \sqrt[9]{(-3)^3} = (-3)^{3/9} \).
03

Simplify the exponent

Simplify the exponent \( \frac{3}{9} \), which reduces to \( \frac{1}{3} \). Thus the expression \( (-3)^{3/9} \) simplifies to \( (-3)^{1/3} \).
04

Simplify radical form

The expression \( (-3)^{1/3} \) is equivalent to \( \sqrt[3]{-3} \). Which is the simplest radical form without imaginary numbers.

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Key Concepts

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

Rationalizing the Denominator
Rationalizing the denominator is a process used to eliminate radicals from the denominator in a fraction. This technique is essential because having radicals in the denominator is not considered proper mathematical form. A rationalized denominator is one without a radical. Here's a step-by-step approach:
  • Identify the radical in the denominator.
  • Multiply both the numerator and the denominator by the conjugate if the denominator is a binomial (such as \( \sqrt{a} + \sqrt{b} \)), or simply the radical itself if it is a monomial.
  • Apply the distributive property to combine and simplify the terms.
  • Simplify the fraction, ensuring there are no radicals in the denominator.
Understanding how and why we rationalize the denominator is crucial in mathematics, as it helps us to manipulate and simplify expressions more easily.
Properties of Exponents
Exponents are powerful tools that allow us to write repeated multiplication in a compact form. They obey several important properties that make simplifying expressions easier. Some key properties of exponents include:
  • Product of Powers: \( a^m \cdot a^n = a^{m+n} \)
  • Power of a Power: \( (a^m)^n = a^{mn} \)
  • Power of a Product: \( (ab)^m = a^m \cdot b^m \)
  • Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \) (provided \( a eq 0 \))
  • Negative Exponent: \( a^{-m} = \frac{1}{a^m} \)
  • Zero Exponent: \( a^0 = 1 \) (assuming \( a eq 0 \))
When solving problems involving radicals and exponents, these properties help us simplify complex expressions systematically. For example, in the problem \( \sqrt[9]{(-3)^3} = (-3)^{3/9} = (-3)^{1/3} \), we used the Power of a Power property to convert the root to fractional exponent form.
Radical Expressions
Radical expressions involve roots, such as square roots, cube roots, and other types of roots, denoted by the radical symbol \( \sqrt{} \). Simplifying radical expressions often involves expressing numbers or variables under a radical sign in their simplest form.For example, to simplify \( \sqrt[9]{-27} \), we recognize that \( -27 \) can be written as \( (-3)^3 \). Representing this under a root using properties of exponents, \( \sqrt[9]{(-3)^3} \) becomes \( (-3)^{3/9} \), further simplifying to \( \sqrt[3]{-3} \), which is the simplest radical form.
  • Identifying Perfect Powers: Break down numbers into factors to find if a root can be simplified.
  • Converting Radicals to Exponents: Using \( a^{m/n} = \sqrt[n]{a^m} \) can make certain calculations more straightforward.
  • Combining Radicals: Radicals can be combined through multiplication and division if they have the same index.
These strategies and insights into radical expressions help in mastering simplification techniques vital for algebra and calculus.

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Most popular questions from this chapter

Perform the indicated operations, expressing answers in simplest form with rationalized denominators. $$(\sqrt{\frac{2}{R}}+\sqrt{\frac{R}{2}})(\sqrt{\frac{2}{R}}-2 \sqrt{\frac{R}{2}})$$

Express each of the given expressions in simplest form with only positive exponents. $$\frac{6^{-1}}{4^{-2}+2}$$

$$\text {Solve the given problems.}$$ $$\begin{aligned} &\text { Evaluate } r^{2}-s^{2} \text { if } r=-b+\sqrt{b^{2}-4 a c} \text { and }\\\ &s=-b-\sqrt{b^{2}-4 a c} \end{aligned}$$

Perform the indicated operations. A factor used in measuring the loudness sensed by the human ear is \(\left(I / I_{0}\right)^{0.3},\) where \(I\) is the intensity of the sound and \(I_{0}\) is a reference intensity. Evaluate this factor for \(I=3.2 \times 10^{-6} \mathrm{W} / \mathrm{m}^{2}\) (ordinary conversation) and \(I_{0}=10^{-12} \mathrm{W} / \mathrm{m}^{2}\)

Rationalize the numerator of each fraction. $$\frac{\sqrt{19}-3}{5}$$
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