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Chapter 7: Problem 8

Use radical notation to write each expression. Simplify if possible. $$ 81^{1 / 4} $$

Short Answer

Expert verified
The expression is \( \sqrt[4]{81} = 3 \).

Step by step solution

01

Understand the Expression

The given expression is \(81^{1/4}\). This is written in exponential form where 81 is the base and \(\frac{1}{4}\) is the exponent.
02

Convert to Radical Notation

In radical notation, \(a^{\frac{1}{n}}\) can be expressed as \(\sqrt[n]{a}\). Therefore, \(81^{1/4}\) can be written as \(\sqrt[4]{81}\).
03

Simplify the Radical Expression

The expression \(\sqrt[4]{81}\) is asking what number, when multiplied by itself four times, equals 81. Recognize that 81 is \(3^4\) because \(3 \times 3 \times 3 \times 3 = 81\). Hence, \(\sqrt[4]{81} = 3\).

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Key Concepts

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

Exponential Form
Exponential form allows us to express a number as a base raised to a power. This is particularly useful in simplifying and calculating large powers and roots. For example, the expression \(81^{1/4}\) is in exponential form where 81 is the base and \(\frac{1}{4}\) is the exponent.
The base represents the number we will be multiplying, while the exponent tells us how many times we will multiply this number.
* Key Characteristics of Exponential Form:
  • Provides a concise way to express repeated multiplication.
  • Uses a base number and an exponent (which can be a fraction, whole number, or negative number).
  • Helps in converting numbers to or from scientific and polar notation.
When dealing with fractional exponents like \(\frac{1}{4},\) it indicates a root, and this bridges us into radical notation, making the transition between exponents and roots seamless.
Simplifying Radicals
Simplifying radicals involves finding a simpler or more exact expression for a given radical. This often means factoring the number underneath the radical to find a perfect power. Let's consider our original problem \(\sqrt[4]{81}\).
This expression is asking for the number that, when raised to the fourth power, equals 81. Identifying that 81 can be rewritten as a power of 3, specifically \(3^4\), is the key.
* Steps to Simplify Radicals:
  • Identify the prime factorization or perfect power of the number within the radical. For 81, this is \(3^4\).
  • Use the property \(\sqrt[n]{a^n} = a\) where \(a\) is the base.
  • If the expression equals the radical's index, extract the base (e.g., \(\sqrt[4]{81} = 3\) because \(81 = 3^4\)).
Simplifying helps reduce complex expressions to simpler, easily understandable forms. It also aids in further calculations, ensuring precision and ease of computation in advanced mathematics.
Fourth Root
The fourth root of a number is a special case of radical expressions where you're looking for a number that when multiplied by itself four times gives the original number. Think of it as unwinding or un-multiplying the number back to its base.
When we talk about \(\sqrt[4]{81}\), we're searching for what number \(x\) satisfies \(x^4 = 81\). Since \(81 = 3 \times 3 \times 3 \times 3\), the fourth root is readily found to be 3.
Here's a quick overview:
  • Fourth roots can also be expressed in exponential form as \(a^{1/4}\).
  • This concept is crucial when dealing with equations involving higher powers and roots.
  • It involves understanding both exponential growth and reduction.
The process of finding the fourth root reaffirms the power of understanding how numbers and their powers interact, reinforcing their roles in diverse mathematical contexts such as geometry, algebra, and calculus.

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Most popular questions from this chapter

The Mosteller formula for calculating adult body surface area is \(B=\sqrt{\frac{h w}{3131}},\) where \(B\) is an individual's body surface area in square meters, \(h\) is the individual's height in inches, and \(w\) is the individual's weight in pounds. Use this information to answer Exercises 113 and 114 . Round answers to 2 decimal places. Find the body surface area of an individual who is 74 inches tall and who weighs 225 pounds.

Simplify. Assume that variables represent positive real numbers. $$ \sqrt{x^{10}} $$

Determine whether the following are real numbers. $$ \sqrt{-17} $$

Find each root. Assume that all variables represent nonnegative real numbers. $$ \sqrt[4]{81 x^{4}} $$

Simplify. Assume that the variables represent any real number. $$ \sqrt{(x-5)^{2}} $$
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