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Chapter 10: Problem 19

Multiply. $$ \sqrt[3]{0.7 y} \sqrt[3]{0.3 y} $$

Short Answer

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Step by step solution

01

Apply the Property of Radicals

Use the property of cube roots: \ \ \ \( \sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b} \ \ \ \). In this case, \ \ \( \sqrt[3]{0.7 y} \cdot \sqrt[3]{0.3 y} = \sqrt[3]{(0.7 y) \cdot (0.3 y)} \ \ \ \).
02

Multiply the Expressions Inside the Radical

Perform the multiplication \ \ \( (0.7 y) \cdot (0.3 y) \ \ \ \). \ \ \This results in \ \ \( 0.7 \cdot 0.3 \cdot y \cdot y = 0.21 y^2 \ \ \ \).
03

Simplify the Expression

Now, take the cube root of the product: \ \ \( \sqrt[3]{0.21 y^2} \ \ \ \). This is already in its simplified form as there are no perfect cubes to further simplify.

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Key Concepts

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

Properties of Radicals
When working with radicals, there are several important properties to keep in mind. A radical can be a square root, cube root, or any root. For cube roots specifically, one useful property is that the product of cube roots is equal to the cube root of the product. For example, \( \sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \:\cdot \: b} \). This property simplifies problems involving the multiplication of radical expressions, as it allows you to combine the numbers under a single radical.
Multiplying Expressions
To multiply expressions under a radical, such as cube roots, you first apply the property we discussed. For example, \ \( \sqrt[3]{0.7 y} \cdot \sqrt[3]{0.3 y} \ \) becomes \ \(\sqrt[3]{(0.7 y) \cdot (0.3 y)} \ \). Next, you multiply the numbers and variables inside the radical separately:
  • Numbers: Multiply 0.7 by 0.3 to get 0.21.
  • Variables: Multiply \ \( y \ \times \ \ y = y^2 \ \).
This results in \ \( \sqrt[3]{0.21 y^2} \ \).
Simplifying Radicals
Simplifying radicals means expressing the radical in its most reduced form. For cube roots, this involves looking for any perfect cubes that can be factored out. In the example \ \(\sqrt[3]{0.21 y^2}\ \), 0.21 and \ \( y^2 \ \) are already in their simplest forms because neither contains a cube factor. Therefore, \ \( \sqrt[3]{0.21 y^2} \ \) is already simplified. Always double-check for perfect cubes when simplifying radicals:
  • For numbers, identify cube factors like 8, 27, etc.
  • For variables, look for cube exponents like \ \( a^3 \ \).

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Most popular questions from this chapter

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