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Chapter 0: Problem 20

Simplify each expression. Assume that all variables are positive when they appear. $$ \sqrt[3]{54}$$

Short Answer

Expert verified
3 \sqrt[3]{2}.

Step by step solution

01

Find the Prime Factorization of 54

First, determine the prime factors of 54. 54 can be divided by 2 (since it is even), so we get 54 = 2 * 27. Next, factor 27, which is 27 = 3^3. Therefore, 54 = 2 * 3^3.
02

Simplify the Expression Using Cube Root Properties

Apply the cube root to the prime factorization of 54, using the property \(\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}\). So, we get \(\sqrt[3]{54} = \sqrt[3]{2 \times 3^3} = \sqrt[3]{2} \times \sqrt[3]{3^3}\).
03

Evaluate the Cube Roots

Evaluate \(\sqrt[3]{3^3}\). Since the cube root of a cubed number is the number itself, \(\sqrt[3]{3^3} = 3\). Therefore, \(\sqrt[3]{54} = \sqrt[3]{2} \times 3\).
04

Combine the Simplified Parts

Combine the simplified parts to find the final answer: \(\sqrt[3]{54} = 3 \sqrt[3]{2}\).

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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 expressing a number as the product of its prime factors. Prime factors are numbers that are only divisible by 1 and themselves. For example, to find the prime factorization of 54, we start by dividing 54 by the smallest prime number 2 (since 54 is even):
54 梅 2 = 27
Next, we factor 27. The prime factors of 27 are all 3's (since 3 is a prime number):
27 = 3 脳 3 脳 3 or 3^3.
Combining these results, we get the prime factorization of 54 as:
54 = 2 * 3^3.
Cube Root Properties
Understanding cube root properties is essential for simplifying cube roots. A cube root is written as \( \sqrt[3]{} \) and is the number that, when multiplied by itself three times, gives the original number. Some key properties include:
  • \( \sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b} \)
  • \( \sqrt[3]{a^3} = a \)
These properties help break down complex expressions into simpler components. Using these properties, we can work with each part separately and then combine the results, making the simplification process more manageable.
Simplifying Radicals
Simplifying radicals involves restructuring the expression to its simplest form. When simplifying cube roots, we use the prime factorization and cube root properties:
For example, simplify \( \sqrt[3]{54} \):
Given the prime factorization, 54 = 2 * 3^3, apply the cube root to both parts:
\( \sqrt[3]{54} = \sqrt[3]{2 \times 3^3} \).
Using \( \sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b} \), we get:
\( \sqrt[3]{2 \times 3^3} = \sqrt[3]{2} \times \sqrt[3]{3^3} \).
Since \( \sqrt[3]{3^3} = 3 \), it simplifies to:
\( \sqrt[3]{2} \times 3 \).
Thus, \( \sqrt[3]{54} = 3\sqrt[3]{2} \).
Evaluating Cube Roots
Evaluating cube roots involves finding the number that, when cubed, gives the original number. Here鈥檚 how you can evaluate \( \sqrt[3]{54} \):
After simplifying, we broke it down to \( \sqrt[3]{2} \times \sqrt[3]{3^3} \). Since the cube root of a cubed number is the number itself:
\( \sqrt[3]{3^3} = 3 \).
Thus, it's easier to evaluate real numbers directly from their prime factors:
So, \( \sqrt[3]{54} = \sqrt[3]{2} \times 3 \).
This simplifies to:
\( 3\sqrt[3]{2} \).
By breaking it down into simpler steps, you can handle even complex cube roots efficiently.

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