Problem 14 - Intermediate Algebra

Question

Fill in the blanks. \(\sqrt[3]{x}=y\) if \(y^{3}= \square\).

Step-by-Step Solution

Answer

The blank is filled by \(x\) so \(y^3 = x\).

Step 1: Understand the Problem Statement

We are given the expression \(\sqrt[3]{x}=y\) and need to fill in the blank in the equation \(y^{3} = \square\). This requires knowledge of cube roots and their mathematical properties.

Step 2: Define the Cube Root Property

Recall that if \(\sqrt[3]{x} = y\), then \(y\) is the cube root of \(x\). The cube of \(y\) brings us back to \(x\), since cubing a cube root undoes the cube root.

Step 3: Solve for the Blank

Using the relation from the cube root property, \(y^3 = x\). Therefore, the blank in the equation \(y^3 = \square\) is filled by \(x\).

Key Concepts

Properties of Exponents

Understanding exponents is crucial for tackling cube roots and other mathematical operations. Exponents represent repeated multiplication. The expression \(y^3\) signifies that \(y\) is multiplied by itself three times. This property of exponents is foundational.
It is important to note:

• Any number to the power of one remains unchanged (e.g., \(y^1 = y\)).
• A number to the power of zero equals one (e.g., \(y^0 = 1\)), unless \(y\) is zero.
• Multiplying like bases involves adding their exponents (e.g., \(y^m \cdot y^n = y^{m+n}\)).
• Raising a power to a power means multiplying their exponents (e.g., \((y^m)^n = y^{m \cdot n}\)).

These properties help simplify expressions by allowing us to work effectively with powers.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form. The idea is to make complex equations easier to handle. In the exercise, simplifying starts by understanding the cube root: \(\sqrt[3]{x} = y\).
Here's how you simplify when dealing with cube roots and related expressions:

• Recognize the cube root \(\sqrt[3]{x}\) as the number \(y\) which, when cubed, gives \(x\).
• Apply the inverse operation, which in this case is cubing \(y\), to get back \(x\), thus \(y^3 = x\).
• Simplify complex expressions by following these basic operations, breaking them down step by step.

This process of simplification helps in understanding and solving mathematical problems effectively.

Mathematical Reasoning

Mathematical reasoning involves thinking logically about numbers and operations to solve problems. It's not just about getting the right answer, but understanding why it's correct.
In our exercise, reasoning starts by identifying the relationship between the cube root and the original number. We reason that since \(y\) is the cube root of \(x\), cubing \(y\) must return us to \(x\). This understanding comes from knowing that cube roots and cubes are inverse operations.
Consider these reasoning steps:

• Understand the problem: What does it ask?\( \sqrt[3]{x} = y\) means \(y\) is such that \(y^3 = x\).
• Use known properties: Knowing cube roots, properties of exponents, and algebra helps us deduce the answer.
• Verify the solution: Cross-check if cubing \(y\) indeed gives you \(x\), as expected.

Mathematical reasoning allows for effective problem-solving and aids in applying mathematical concepts correctly.