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Chapter 8: Problem 10

Simplify. $$\sqrt[3]{40}$$

Short Answer

Expert verified
\( \sqrt[3]{40} = 2 \sqrt[3]{5} \)

Step by step solution

01

Factorize the Radicand

Factorize 40 into its prime factors. We have:40 = 2^3 * 5
02

Apply the Cube Root

Apply the cube root to the factorized form. Using properties of cube roots, we get:\( \sqrt[3]{40} = \sqrt[3]{2^3 \cdot 5} \)
03

Simplify the Cube Root

Since \( \sqrt[3]{2^3 \cdot 5} = \sqrt[3]{2^3} \cdot \sqrt[3]{5} \), simplify the cube root of \(2^3\):\( \sqrt[3]{2^3} = 2\).Thus, \( \sqrt[3]{40} = 2 \sqrt[3]{5} \)

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Key Concepts

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

Prime Factorization
To simplify cube roots, we start with prime factorization. This means breaking down a number into its basic building blocks鈥攑rime numbers. For instance, the number 40 can be factorized into prime numbers as follows:
40 = 2^3 * 5
The prime factors here are 2 and 5. Notice how we use exponents to show the repeated factors. Understanding this helps us handle cube roots better.
Properties of Cube Roots
Cube roots have some unique properties that make them different from square roots. One key property is that \( \sqrt[3]{a*b} = \sqrt[3]{a} * \sqrt[3]{b} \).
This means you can take the cube root of each factor separately. Let's apply this to our example:
\( \sqrt[3]{40} \) turns into \( \sqrt[3]{2^3 * 5} \) which then becomes \( \sqrt[3]{2^3} * \sqrt[3]{5} \).
This is a handy property that simplifies calculations.
Simplifying Radicals
Once we apply the properties of cube roots, we can simplify our expression further. Using our earlier factorization, we get:\( \sqrt[3]{2^3 * 5} \) = \( \sqrt[3]{2^3} * \sqrt[3]{5} \)
Since \( \sqrt[3]{2^3} = 2 \), our expression simplifies to 2 \( \sqrt[3]{5} \).
So, \( \sqrt[3]{40} \) in its simplest form is 2 \( \sqrt[3]{5} \). This step-by-step simplification shows us how breaking down numbers and applying radical properties makes complex calculations manageable.

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