Problem 8 - Intermediate Algebra

Question

Fill in the blanks. Consider \(\frac{\sqrt[3]{a}}{\sqrt[3]{x^{2}}}\) and \(\sqrt[3]{\frac{a}{x^{2}}} .\) Which expression is a. the cube root of a quotient? b. the quotient of cube roots? c. How are these two expressions related?

Step-by-Step Solution

Answer

a: \(\sqrt[3]{\frac{a}{x^{2}}}\), b: \(\frac{\sqrt[3]{a}}{\sqrt[3]{x^{2}}}\), c: They are equivalent expressions.

Step 1: Identify the Cube Root of a Quotient

Given the expression \(\sqrt[3]{\frac{a}{x^{2}}}\), identify it as the cube root of a quotient, because the numerator and the denominator are inside a single cube root.

Step 2: Identify the Quotient of Cube Roots

Given the expression \(\frac{\sqrt[3]{a}}{\sqrt[3]{x^{2}}}\), identify it as the quotient of cube roots. Here, the numerator and denominator are separately under a cube root symbol.

Step 3: Compare the Two Expressions

Observe that both expressions represent the same mathematical operation. By the property \(\sqrt[3]{\frac{a}{x^{2}}} = \frac{\sqrt[3]{a}}{\sqrt[3]{x^{2}}}\), these two expressions are equivalent because extracting the cube root of a quotient is the same as taking the cube root of the numerator and the cube root of the denominator separately.

Key Concepts

Cube Root of a Quotient

When you come across an expression like \(\sqrt[3]{\frac{a}{x^2}}\), you are dealing with the cube root of a quotient. This type of expression has both the numerator \(a\) and the denominator \(x^2\) enclosed within the same cube root. Inside the cube root symbol, the division is performed before taking the cube root. Essentially, this means that you first divide \(a\) by \(x^2\) and then find the cube root of the resulting value.

• It is a single operation performed on the entire fraction.
• This expression helps to simplify calculations when dealing with division inside cube roots.

Understanding this concept is crucial as it influences how you perform calculations and affects the order of operations required.

Quotient of Cube Roots

Consider the expression \(\frac{\sqrt[3]{a}}{\sqrt[3]{x^2}}\). This represents the quotient of cube roots. Each part of the fraction – both the numerator \(\sqrt[3]{a}\) and the denominator \(\sqrt[3]{x^2}\) – is separately under its own cube root symbol. Here, you take the cube root of \(a\) and the cube root of \(x^2\) independently, then form a fraction out of the results.

• This means you are performing the cube root on each component before dividing.
• This expression format is especially useful when the individual components are easier to simplify separately.

Using the quotient of cube roots approach sometimes leads to simpler expressions, particularly when simplifying or factoring individual numerators and denominators.

Equivalent Expressions

The expressions \(\sqrt[3]{\frac{a}{x^2}}\) and \(\frac{\sqrt[3]{a}}{\sqrt[3]{x^2}}\) might look different at first glance, but they are actually equivalent. The equivalency is grounded in a fundamental property of cube roots: \(\sqrt[3]{\frac{a}{x^2}} = \frac{\sqrt[3]{a}}{\sqrt[3]{x^2}}\). This demonstrates that taking the cube root of a fraction is the same as dividing the cube roots of the numerator and denominator separately.

• Understanding this equivalence is key to solving many algebraic problems.
• This set of principles often simplifies complex expressions and verifies their correctness.

Remember that recognizing these equivalent forms can enhance your problem-solving efficiency and provide alternative methods to approach mathematical challenges.