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Chapter 7: Problem 59

Simplify. \(\sqrt[3]{40}\)

Short Answer

Expert verified
2 \times \sqrt[3]{5}

Step by step solution

01

Express 40 as a Product of Prime Factors

First, express the number 40 as a product of its prime factors. We know that 40 = 2^3 * 5.
02

Apply the Cube Root to Each Factor

Next, apply the cube root to each factor separately. This gives us \ \ \[ \sqrt[3]{2^3 \times 5} = \sqrt[3]{2^3} \times \sqrt[3]{5} \]
03

Simplify the Cube Roots

Now, simplify the cube roots. The cube root of 2^3 is 2, because \[ \sqrt[3]{2^3} = 2 \]. Thus, the expression simplifies to \ \[ 2 \times \sqrt[3]{5} \]

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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 composite number into its prime factors. These are the prime numbers that, when multiplied together, give back the original number. For instance, to factorize 40, we divide it by the smallest prime number (2) until we can't anymore. We get:
  • 40 梅 2 = 20
  • 20 梅 2 = 10
  • 10 梅 2 = 5 (5 is a prime number)
So, 40 can be expressed as 2 脳 2 脳 2 脳 5 or 2^3 脳 5. Knowing this is foundational for simplifying cube roots!
Cube Root
A cube root finds a number which when multiplied by itself three times gives the original number. The cube root of a number n is written as \(\sqrt[3]{n} \). For example, \(\sqrt[3]{27} \) is 3 because 3 脳 3 脳 3 = 27. When you face a cube root involving a product of prime factors, you can split the cube root across each factor. For instance, to simplify \(\sqrt[3]{40} \), we use prime factorization to rewrite 40 as 2^3 脳 5 and then split the root:
\[\sqrt[3]{2^3 \times 5} = \sqrt[3]{2^3} \times \sqrt[3]{5} \]. The cube root of 2^3 is straightforward (since \(\sqrt[3]{2^3} = 2 \)), making our expression easier to handle.
Simplifying Expressions
Simplifying expressions means making them easier to work with or understand. We do this by breaking down components and combining like terms. For cube roots, simplifying involves two key parts:
  • Prime factorization: Rewriting the number under the root in terms of its prime factors.
  • Applying the root: Simplifying the root for those individual factors.
Using our example, \(\sqrt[3]{40}\rightarrow 2 \times \sqrt[3]{5} \). The simplified form tells us that while the factor of 2 comes out as an integer, the root \(\sqrt[3]{5}\rightarrow \) stays as a cube root expression. This process turns a more complex expression into an easier form to work with.

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