Problem 19 Multiply. $$\sqrt[3]{0.7 y} \s... [FREE SOLUTION]

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Intermediate Algebra : Concepts and Applications

Marvin L. Bittinger, David J. Ellenbogen, Barbara L. Johnson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 7: Problem 19

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

Short Answer

Expert verified

$\sqrt[3]{0.21 y^2}$

Step by step solution

Identify the Expression

The given expression to multiply is $\sqrt[3]{0.7 y} \sqrt[3]{0.3 y}$. Notice both terms are cube roots.

Use the Property of Cube Roots

Recall that $\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}$. Apply this property to combine the expressions.

Multiply the Values Inside the Cube Roots

Multiply the values inside the cube roots: $0.7y \cdot 0.3y = 0.21y^2$.

Simplify the Result

Combine the product inside a single cube root: $\sqrt[3]{0.21 y^2}$. This is the simplified result of the multiplication.

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Key Concepts

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

Cube Roots

A cube root is a number that, when multiplied by itself three times, yields the original number.
In mathematical terms, the cube root of a number 'a' is represented as $ \sqrt[3]{a} $.
For example, $ \sqrt[3]{8} = 2 $ because $ 2 \times 2 \times 2 = 8 $.
Cube roots can be extracted from both simple numbers and algebraic expressions.
In our specific problem, we have cube roots of expressions involving both numbers and variables: $ \sqrt[3]{0.7y} $ and $ \sqrt[3]{0.3y} $.
Learning to handle cube roots is vital for simplifying and solving algebraic expressions involving roots.

Multiplication

Multiplying cube roots follows some unique properties.
One of the most important properties is that the product of two cube roots is equal to the cube root of the product of the numbers under the roots.
Mathematically, this is shown as: $ \sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{a \times b} $.
Using this property makes it easier to handle and simplify expressions.
In the exercise, we multiply $ \sqrt[3]{0.7y} $ with $ \sqrt[3]{0.3y} $ using the property above.

Simplification of Expressions

Simplification is the process of making an expression easier to work with.
When simplifying expressions involving cube roots, you combine and reduce terms as much as possible.
In our exercise, after applying the multiplication property of cube roots, we need to multiply $ 0.7y $ and $ 0.3y $.
We get: $ 0.7 \times 0.3 = 0.21 $ and $ y \times y = y^2 $.
Hence, $ 0.7y \times 0.3y = 0.21y^2 $.
The final simplified form is then written as $ \sqrt[3]{0.21y^2} $.
Simplifying expressions this way ensures they are in their most manageable and easily understood form.

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91影视

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

Short Answer

Step by step solution

Identify the Expression

Use the Property of Cube Roots

Multiply the Values Inside the Cube Roots

Simplify the Result

Key Concepts

Cube Roots

Multiplication

Simplification of Expressions

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