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Chapter 7: Problem 10

Multiply. $$\sqrt[3]{2} \sqrt[3]{3}$$

Short Answer

Expert verified
\[ \sqrt[3]{2} \times \sqrt[3]{3} = \sqrt[3]{6} \]

Step by step solution

01

Understand the Problem

The task is to multiply two cube roots: \(\sqrt[3]{2} \) and \(\sqrt[3]{3}\).
02

Use the Property of Cube Roots

Recall the property of cube roots that states: \(\sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{a \times b}\).
03

Apply the Property

Apply this property to the given expressions: \(\sqrt[3]{2} \times \sqrt[3]{3} = \sqrt[3]{2 \times 3}\).
04

Simplify Inside the Cube Root

Multiply the numbers inside the cube root: \(\sqrt[3]{2 \times 3} = \sqrt[3]{6}\).

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Key Concepts

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

Properties of Radicals
Radicals, like cube roots, have important properties that simplify operations like multiplication and division. One key property is that the product of two cube roots is the cube root of the product of the numbers inside the roots. For example, the multiplication property for cube roots is: \( \sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{a \times b} \). Another useful property is that radicals can be simplified if the number inside the root is a perfect power of the root's index. These properties help in reshaping complex radical expressions into simpler, more manageable forms.
Simplifying Radicals
When simplifying radicals, the goal is to make the expression under the root sign as simple as possible. For cube roots, we look for factors that are perfect cubes. If we identify any, we can simplify the cube root accordingly. For example, \( \sqrt[3]{8} = 2 \) because \( 8 = 2^3 \). In cases without perfect cubes, we leave the expression in its simplest radical form. Consider \( \sqrt[3]{6} \. \) Since 6 isn't a perfect cube and has no perfect cube factors, the expression remains \( \sqrt[3]{6} \).
Cube Roots
Cube roots are one of the most common types of radicals. They are used to find a number which, when multiplied by itself three times, gives the original number. Symbolically, the cube root of a number a is written as \( \sqrt[3]{a} \. \). For example, \( \sqrt[3]{8} = 2 \) because \( 2 \times 2 \times 2 = 8 \). These roots are particularly useful when dealing with volume calculations, as they can reverse the operation of cubing a number. Additionally, cube roots follow specific rules, like the property for multiplication described earlier. Understanding and using these properties make working with radicals much easier.

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Most popular questions from this chapter

The absolute value of a complex number \(a+b i\) is its distance from the origin. (See the graph above.) Using the distance formula, we have \(|a+b i|=\sqrt{a^{2}+b^{2}}\) Find the absolute value of each complex number. $$|8-6 i|$$

Find the midpoint of the segment with the given endpoints. $$ \left(-\frac{4}{5},-\frac{2}{3}\right) \text { and }\left(\frac{1}{8}, \frac{3}{4}\right) $$

Factor completely. $$12 a^{3}-5 a^{2}-3 a$$

Let \(f(x)=x^{2} .\) Find each of the following. $$ f(\sqrt{2}+\sqrt{10}) $$

Simplify. $$ 7 x \sqrt{(x+y)^{3}}-5 x y \sqrt{x+y}-2 y \sqrt{(x+y)^{3}} $$
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