/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 97 \(\sqrt[4]{81}\)... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 97

\(\sqrt[4]{81}\)

Short Answer

Expert verified
The fourth root of 81 is 3.

Step by step solution

01

Understand the problem

The problem is asking for the fourth root of 81, which means we need to find a number that when raised to the power of 4 equals 81.
02

Express 81 as a power of 3

Notice that 81 can be written as a power of 3: \[ 81 = 3^4 \]
03

Use the property of roots

The fourth root of a number can be found by taking the number to the power of 1/4. This gives us: \[ \sqrt[4]{81} = (3^4)^{1/4} \]
04

Simplify the expression

Simplify the expression using the property that \(a^{m/n} = (a^m)^{1/n}\), so: \[ (3^4)^{1/4} = 3 \]

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Key Concepts

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

radical expressions
A radical expression involves roots, such as square roots, cube roots, and so on. In our case, we're dealing with the fourth root, which means we're looking for a number that, when raised to the power of 4, equals 81. Understanding how to manipulate roots and recognize patterns can be very helpful. Radical expressions are represented with the radical symbol (鈭�) and can sometimes be simplified or converted to an expression with exponents. For example, finding \( \sqrt[4]{81} \) involves recognizing that it can be rewritten using properties of exponents.
exponents
Exponents are a way to express repeated multiplication of a number by itself. For instance, 3 raised to the 4th power (\( 3^4 \)) means \(3 \times 3 \times 3 \times 3 = 81\). In our problem, we needed to find what number raised to the 4th power gives us 81. Exponents follow specific rules that simplify their manipulation. One relevant rule for our problem is that \( a^{m/n} = (a^m)^{1/n}\). This rule helps us convert radical expressions into simpler forms that are easier to understand and calculate.
simplifying radicals
Simplifying radicals is all about making the expression as simple as possible. When simplifying \( \sqrt[4]{81} \), we start by recognizing that 81 can be expressed as \( 3^4 \). This step is crucial because it converts our radical into a more easily manageable form. Next, we use properties of exponents to simplify further: \( \sqrt[4]{81} = (3^4)^{1/4} \). Simplifying \((3^4)^{1/4}\) by multiplying the exponents, \((4 \times \frac{1}{4})\), we get 3. So, we've simplified \( \sqrt[4]{81} \) to 3. Recognizing patterns and properties in exponents and radicals can make this process much more straightforward.

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Most popular questions from this chapter

The windchill factor is a measure of the cooling effect that the wind has on a person's skin. It calculates the equivalent cooling temperature if there were no wind. The National Weather Service uses the formula $$ \text { Windchill temperature }=35.74+0.6215 T-35.75 V^{4 / 25}+0.4275 T V^{4 / 25} $$ where T is the temperature in \(^{\circ} \mathrm{F}\) and \(\mathrm{Vi}\) is the wind speed in miles per hour, to calculate wind- chill. The table gives the windchill factor for various wind speeds and temperatures at which frostbite is a risk, and how quickly it may occur. Use the formula to determine the windchill temperature to the nearest tenth of a degree, given the following conditions. Compare answers with the appropriate entries in the table. \(40^{\circ} \mathrm{F}, 10\) -mph wind

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