/*! 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 57 Simplify \(\sqrt[3]{128}\)... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 57

Simplify \(\sqrt[3]{128}\)

Short Answer

Expert verified
The simplified form of \[\textbackslash\( \textbackslash\sqrt[3]\textbackslash{128}\displaystyle\]

Step by step solution

01

Understand the Problem

We need to find the cube root of 128.
02

Prime Factorization

First, find the prime factorization of 128. 128 is even, so divide by 2:128 梅 2 = 6464 梅 2 = 3232 梅 2 = 1616 梅 2 = 88 梅 2 = 44 梅 2 = 22 梅 2 = 1Thus, the prime factorization of 128 is:128 = 2^7
03

Simplify Using Cube Roots

We need to take the cube root of 128, which is equivalent to taking the cube root of each factor in its prime factorized form.Cube root of 128 is ewlineewlineewline ewline ewline ewlineewline ewlineewline ewlineso take the cube root of each factor individually:ewline=ewline=The cube root of 2^6 is 2^2, and the cube root of 2^1 is approximately equal to 1.26 rounded to two decimal places. Hence the solution is = 4.145.

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Key Concepts

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

Prime Factorization
In this exercise, our goal is to simplify \( \sqrt[3]{128} \). Prime factorization is a fundamental step.
It involves breaking down 128 into its prime factors. Prime numbers are those that have only two distinct positive divisors: 1 and the number itself.
Here鈥檚 how we factorize 128:

  • 128 is even, so divide by 2:
  • 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
Once we reach 1, we stop. This gives us the prime factorization:
128 = 2^7
Breaking down the number helps illustrate its structure, which we will use in the next steps.
Simplification
Once we have the prime factorization (2^7), we simplify \( \sqrt[3]{128} \). Taking the cube root involves dividing the exponent by 3.
In simpler terms, we are checking how many times 3 fits into the exponent of 7:
2^7 is expressed as 2^{3 \cdot 2 + 1} = 2^6 \cdot 2^1
The trick here is to remember that taking the cube root (鲁鈭�) of something means raising it to the power of \( \frac{1}{3} \)
  • Cube root of 2^{6} = 2^{2} = 4
  • Cube root of 2^1 = \(2^{ \frac{1}{3}} \) (which is approximately 1.26)
Multiplying these together gives us the simplified form:
4 * 1.26 鈮� 5.04
Understanding each part of the exponent is key in simplifying complex numbers.
Radicals in Algebra
Radicals are expressions that contain a root symbol (鈭�). In algebra, understanding how to manipulate them is essential.
For cube roots, we're specifically working with radicals like \( \sqrt[3]{x} \).
Here are key points to remember:

  • The cube root of x means finding a number that, when multiplied by itself three times, equals x.
  • Prime factorization helps break down numbers into more manageable parts.
  • Simplifying radicals involves rules for exponents.
For example, \( \sqrt[3]{2^7} \) can be split into smaller, simpler terms, and each part's cube root is taken individually.
This helps us manage and simplify more complex equations. A deeper understanding of radicals makes advanced algebra easier!

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Most popular questions from this chapter

Write each radical as an exponential and simplify. Leave answers in exponential form. Assume that all variables represent positive numbers. $$ \sqrt[3]{\sqrt{k}} $$

Solve each problem. Meteorologists can determine the duration of a storm using the function $$ T(d)=0.07 d^{3 / 2} $$ where \(d\) is the diameter of the storm in miles and \(T\) is the time in hours. Find the duration of a storm with a diameter of \(16 \mathrm{mi}\). Round the answer to the nearest tenth of an hour.

Work each problem. Suppose someone claims that \(\sqrt[n]{a^{n}+b^{n}}\) must equal \(a+b,\) because when \(a=1\) and \(b=0,\) a true statement results: $$ \sqrt[n]{a^{n}+b^{n}}=\sqrt[n]{1^{n}+0^{n}}=\sqrt[n]{1^{n}}=1=1+0=a+b $$ Explain why this is faulty reasoning.

Write each radical as an exponential and simplify. Leave answers in exponential form. Assume that all variables represent positive numbers. $$ \sqrt[3]{\sqrt[5]{\sqrt{y}}} $$

Simplify. Assume that all variables represent positive real numbers. \(\sqrt[3]{4} \cdot \sqrt{3}\)
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