Problem 50 - Elementary and Intermediate Algebra

Question

Square or cube each quantity and simplify the result. $$ (-3 \sqrt[3]{10 y^{3}})^{3} $$

Step-by-Step Solution

Answer

-270y^3.

Step 1: Understand the Expression

We're given the expression \[(-3 \sqrt[3]{10 y^3})^3\]Our task is to expand it fully by applying the cube to each part of the expression inside the parentheses.

Step 2: Distribute the Cube Across the Terms

The expression is made of two parts: \(-3\) and \(\sqrt[3]{10 y^3}\). We can apply the cube to each part separately. This gives us: \[(-3)^3 \times (\sqrt[3]{10 y^3})^3\].

Step 3: Cube the Constants

Calculate \((-3)^3\). The expression becomes: \((-3) \times (-3) \times (-3) = -27\). So the constant-part of our expression is \(-27\).

Step 4: Simplify the Cube Root Component

Simplify \((\sqrt[3]{10 y^3})^3\). By the property of cube roots, \((\sqrt[3]{a})^3 = a\). Thus, our expression becomes simply \(10y^3\).

Step 5: Combine the Results

Now combine the results from Steps 3 and 4. Our expression becomes: \(-27 \times 10y^3 = -270y^3.\)

Step 6: Final Solution

The simplified expression is \(-270y^3.\)

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, called the base, to the power of an exponent. The expression \(a^n\) denotes that the base \(a\) is multiplied by itself \(n\) times. In our problem, we started with \[(-3\sqrt[3]{10y^3})^3\].

• The base is \(-3\sqrt[3]{10y^3}\).
• The exponent is 3, meaning we multiply the base by itself three times.

Exponentiation simplifies the process of repetitive multiplication.For constants, raising a number like \(-3\) to the power of 3 is straightforward: \((-3) imes (-3) imes (-3) = -27\).For more complex bases that include variables or other expressions, each component within the base must also be addressed according to the rules of exponentiation.

Cube Roots

Cube roots, denoted as \(\sqrt[3]{a}\), represent a value that, when multiplied by itself three times, gives the original number \(a\). It's the inverse operation of cubing a number.In our expression, \(\sqrt[3]{10y^3}\) is taken.The relationship with cubing is key: the cube of a cube root brings us back to the original number.

• Cubing \(\sqrt[3]{10y^3}\) results in \((\sqrt[3]{10y^3})^3 = 10y^3\).

This demonstrates that applying the cube to a cube root essentially cancels the root, resulting in just the number under the root sign, helping simplify expressions.

Simplification

Simplification is the process of making a mathematical expression as concise and straightforward as possible. After we expanded our original expression \[(-3\sqrt[3]{10y^3})^3\],we needed to simplify it:

• First, separate constants and variable components for individual simplification.
• Cubed constant: \((-3)^3 = -27\).
• Cubed cube root: \((\sqrt[3]{10y^3})^3 = 10y^3\).
• Finally, combine:\(-27 \times 10y^3 = -270y^3\).

Simplification helps reduce complexity and clarify the structure of mathematical expressions, often making further calculations easier.

Mathematical Operations

Mathematical operations are fundamental processes used for solving problems, including addition, subtraction, multiplication, division, and exponentiation. Each operation has specific rules that dictate how expressions are manipulated.In this problem, we mainly focused on multiplication and exponentiation.

• Multiplication is used to combine terms: both constants and variable terms are multiplied.
• Exponentiation was used to address the powers in the expression.
• By multiplying the results of the cubed constant and cubed cube root:\(-27 \times 10y^3 = -270y^3\).

Understanding these operations and their interaction helps in solving complex algebraic expressions efficiently.