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Chapter 8: Problem 53

Express each radical in simplified form. $$ -\sqrt[4]{512} $$

Short Answer

Expert verified
-4\sqrt[4]{2}

Step by step solution

01

Factorize the Radicand

First, factorize 512 into its prime factors. Since 512 is an even number, start by dividing by 2 repeatedly.\begin{align*} 512 \div 2 &= 256 \ 256 \div 2 &= 128 \ 128 \div 2 &= 64 \ 64 \div 2 &= 32 \ 32 \div 2 &= 16 \ 16 \div 2 &= 8 \ 8 \div 2 &= 4 \ 4 \div 2 &= 2 \ 2 \div 2 &= 1 \end{align*}So, 512 can be written as \(2^9\).
02

Apply the Fourth Root

Use the property of radicals that \(\frac{m}{n}\text{鈭歛} = a^{m/n}\). Therefore, \(-\sqrt[4]{512}\) can be rewritten using the prime factorization:\[-\sqrt[4]{2^9} = -\left(2^9\right)^{1/4}\]
03

Simplify the Exponent

Simplify the exponent by multiplying it out: \[ -\left(2^{9/4}\right) \]}Since \(9/4\) can be broken into \(2+1/4\):\[ 2^{2+\frac{1}{4}} = 2^2 \cdot 2^{1/4} \] Finally, simplify it further: \[ -4\sqrt[4]{2} = -4\sqrt[4]{2} \]

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Key Concepts

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

Prime Factorization
Prime factorization is the process of expressing a number as the product of its prime factors. For example, if we take the number 512, we factorize it by continually dividing by the smallest prime number, which is 2 in this case.
We start with:
  • 512 梅 2 = 256
  • 256 梅 2 = 128
  • 128 梅 2 = 64
  • 64 梅 2 = 32
  • 32 梅 2 = 16
  • 16 梅 2 = 8
  • 8 梅 2 = 4
  • 4 梅 2 = 2
  • 2 梅 2 = 1
In the end, we have determined that 512 can be written as the product of nine 2s, or in exponential form, as \(2^9\).
This process simplifies complex calculations and is the foundation for further mathematical operations.
Exponent Rules
Exponent rules are essential to simplifying expressions involving powers and roots. Here, we're particularly interested in a rule that helps us simplify radicals: \[x^{m/n} = \sqrt[n]{x^m}\].
When we deal with the fourth root \(\sqrt[4]{2^9}\), we convert it into exponential form as \((2^9)^{1/4}\).
Using the property of exponents, we can then rewrite the expression as \(2^{9/4}\).
Fourth Roots
A fourth root finds a number that, when multiplied by itself four times, gives the original number. The fourth root of 512, with the expression properly simplified, involves breaking down \(2^{9/4}\).
Recognize that \(9/4\) can be separated into \(2 + 1/4\):
\(2^{9/4} = 2^{2 + 1/4} = 2^2 \cdot 2^{1/4}\).
Since \(2^2 = 4\), our expression becomes:\(-4\sqrt[4]{2}\).
Therefore, \(-\sqrt[4]{512}\) can be simplified entirely as \(-4\sqrt[4]{2}\).
Simplifying expressions by breaking them down into understandable parts helps to unveil the underlying simplicity of seemingly complex radicals.

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Most popular questions from this chapter

In the study of sound, one version of the law of tensions is $$ f_{1}=f_{2} \sqrt{\frac{F_{1}}{F_{2}}} $$ If \(F_{1}=300, F_{2}=60,\) and \(f_{2}=260,\) find \(f_{1}\) to the nearest unit.

Find the distance between each pair of points. $$ (\sqrt{2}, \sqrt{6}) \text { and }(-2 \sqrt{2}, 4 \sqrt{6}) $$

Find each quotient. $$ \frac{2}{1+i} $$

Solve each problem. The following letter appeared in the column "Ask Tom Why,"," written by Tom Skilling of the Chicago Tribune: Dear Tom, I cannot remember the formula to calculate the distance to the horizon. I have a stunning view from my 14 th-floor condo, 150 ft above the ground. How far can I see? Ted Fleischaker; Indianapolis, Ind. Skilling's answer was as follows: To find the distance to the horizon in miles, take the square root of the height of your view in feet and multiply that result by 1.224 . Your answer will be the number of miles to the horizon. Assuming that Ted's eyes are \(6 \mathrm{ft}\) above the ground, the total height from the ground is \(150+6=156 \mathrm{ft}\). To the nearest tenth of a mile, how far can he see to the horizon?

Find each quotient. $$ \frac{8 i}{2+2 i} $$
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