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Chapter 7: Problem 57

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

Short Answer

Expert verified
4 \(\root[3]{2}\)

Step by step solution

01

Prime Factorization

The first step in simplifying \(\root[3]{128}\) is to perform the prime factorization of 128. Start by dividing 128 by 2 (the smallest prime number) to get 64. Continue dividing by 2: \(128 = 2 \times 64\), \(64 = 2 \times 32\), \(32 = 2 \times 16\), \(16 = 2 \times 8\), \(8 = 2 \times 4\), \(4 = 2 \times 2\). Thus, 128 can be written as \128 = 2^7\.
02

Use Cube Root Property

Next, use the cube root property on the prime factorization result. Recall that \(\root[3]{a^3} = a\). We have: \(\root[3]{128} = \root[3]{2^7}\).
03

Simplify Using Properties of Exponents

Separate the exponents in a form that includes cubes. Since \2^7\ can be separated into \2^6 \times 2\, rewrite \(\root[3]{2^7} = \root[3]{2^6 \times 2} = \root[3]{(2^2)^3 \times 2} \).
04

Simplify the Expression

Simplify the expression using cube root properties: \(\root[3]{(2^2)^3 \times 2} = \root[3]{(2^2)^3} \times \root[3]{2} = 2^2 \times \root[3]{2} = 4 \times \root[3]{2} \).

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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 breaking down a number into its prime components. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. For example, when simplifying \(\root[3]{128}\), we start by finding its prime factors.

We divide 128 by the smallest prime number, 2, which results in 64. We keep dividing by 2 until we get to the prime factors:
  • 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
This shows that 128 can be written as \(2^7\). Breaking down the number into its prime factors is crucial for simplifying radical expressions.
Properties of Exponents
Understanding the properties of exponents makes simplifying expressions more manageable. Here are a few essential properties:
  • Product of Powers: \(a^m \times a^n = a^{m+n}\)
  • Power of a Power: \((a^m)^n = a^{m \times n}\)
  • Power of a Product: \((ab)^n = a^n \times b^n\)
In the given example, we use the power of a product property to simplify \(\root[3]{2^7}\). We separate \(2^7\) into \(2^6 \times 2\) and then rewrite \(2^6\) as \((2^2)^3\). This allows us to use the cube root property effectively. Remember, combining these properties is key to simplifying complex expressions.
Cube Root Property
The cube root property states \(\root[3]{a^3} = a\). This property helps in simplifying radical expressions.

In our example, we use it on the prime factorized expression of 128, which is \(2^7\). Rewriting \(2^7\) as \((2^2)^3 \times 2\), we can separate and simplify the cube root:

\[\root[3]{(2^2)^3 \times 2} = \root[3]{(2^2)^3} \times \root[3]{2} = 2^2 \times \root[3]{2} = 4 \times \root[3]{2} \]

Hence, the simplified form of \(\root[3]{128}\) is \(4 \times \root[3]{2}\).

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