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Chapter 7: Problem 40

Simplify the products. Give exact answers. $$\sqrt[3]{5} \cdot \sqrt[3]{100}$$

Short Answer

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

Step by step solution

01

- Express in Radical Form

The given expression is \(\sqrt[3]{5} \cdot \sqrt[3]{100}\). Both terms are cube roots. To simplify the product, you can combine the numbers under one radical sign.
02

- Combine Under Single Radical Sign

Combine the two cube roots: \(\sqrt[3]{5} \cdot \sqrt[3]{100} = \sqrt[3]{5 \cdot 100}\).
03

- Multiply the Numbers Inside the Radical

Next, multiply 5 and 100: \(5 \cdot 100 = 500\). So, \(\sqrt[3]{5 \cdot 100} = \sqrt[3]{500}\).
04

- Simplify the Radical if Possible

Check if 500 can be simplified further. Since 500 can be factored as \(5^3 \cdot 2\), it can be simplified to \(\sqrt[3]{(5^3 \cdot 2)} = 5 \cdot \sqrt[3]{2}\).

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Key Concepts

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

Combining Radicals
When dealing with radicals (like square roots or cube roots), combining them can make the equation easier. In our example, we have \(\sqrt[3]{5} \cdot \sqrt[3]{100}\). Both are cube roots, so we can combine them under a single radical sign. This transforms our equation to \(\sqrt[3]{5 \cdot 100}\). This trick works because the cube root of a product is the same as the product of the cube roots. Just make sure that all radicals are of the same type (e.g., both cube roots).
Multiplying Under a Single Radical
After combining the radicals \(\sqrt[3]{5}\) and \(\sqrt[3]{100}\), the next step is to multiply the numbers inside the radical. This changes our expression: \(\sqrt[3]{5 \cdot 100} = \sqrt[3]{500}\). Multiplying under a single radical simplifies the problem. Only the values inside the radicals are multiplied, while the radical type (cube root in this case) stays the same. This step is useful because it reduces the calculation work in the following steps.
Simplifying Radicals
The final stage of the solution involves simplifying the radical, if possible. We have \(\sqrt[3]{500}\). First, we want to see if the number inside the radical can be factored into a simpler form. Here, 500 factors as \(5^3 \cdot 2\), which simplifies beautifully. So, \(\sqrt[3]{500} = \sqrt[3]{(5^3 \cdot 2)} = 5 \cdot \sqrt[3]{2}\). Simplifying radicals involves breaking down the numbers inside the radical into prime factors or powers and simplifying everything outside the radical. This makes the expression much cleaner and easier to work with.

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Most popular questions from this chapter

Solve each equation. $$(x-3)^{2 / 3}=-4$$

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Discussion Which one of the following expressions is not equivalent to the others? a) \((\sqrt[3]{x})^{4}\) b) \(\sqrt[4]{x^{3}}\) c) \(\sqrt[3]{x^{4}}\) d) \(x^{4 / 3}\) e) \(\left(x^{1 / 3}\right)^{4}\)

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