Question

Challenge Problem Find the difference quotient of the function \(f(x)=\sqrt[3]{x}\) ( Hint: Factor using \(a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right)\) with \(a=\sqrt[3]{x+h}\) and \(b=\sqrt[3]{x} .)\)

Short answer

The difference quotient is \(\frac{1}{\sqrt[3]{x+h}^2 + \sqrt[3]{x+h} \cdot \sqrt[3]{x} + \sqrt[3]{x}^2}\).

Step-by-step solution

- Set up the Difference Quotient

Start by writing down the formula for the difference quotient: \ \[ \frac{f(x+h) - f(x)}{h} \] Given the function \(f(x) = \sqrt[3]{x}\), substitute \(f(x+h)\) and \(f(x)\): \ \[ \frac{\sqrt[3]{x+h} - \sqrt[3]{x}}{h} \]

- Apply the Hint

Use the hint provided. Set \(a = \sqrt[3]{x+h}\) and \(b = \sqrt[3]{x}\). Rewrite the expression using the factoring formula for \(a^3 - b^3\): \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \] Here, \(a^3 = x+h\) and \(b^3 = x\).

- Rewrite the Expression

Rewrite \(\sqrt[3]{x+h} - \sqrt[3]{x}\) in a factored form: \[ \frac{(\sqrt[3]{x+h} - \sqrt[3]{x})}{h} = \frac{a - b}{h} \] Since \(a^3 - b^3 = x+h - x = h\), we can divide both sides by \(h\): \[ \frac{a^3 - b^3}{h} = \frac{h}{h} = 1 \]

- Simplify the Expression

Since \(a^3 - b^3 = h\), we can use the identity: \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \] Thus, \[ \frac{(\sqrt[3]{x+h} - \sqrt[3]{x})(\sqrt[3]{x+h}^2 + \sqrt[3]{x+h} \cdot \sqrt[3]{x} + \sqrt[3]{x}^2)}{h} = 1 \] The result simplifies to: \[ \frac{1}{\sqrt[3]{x+h}^2 + \sqrt[3]{x+h} \cdot \sqrt[3]{x} + \sqrt[3]{x}^2} \]

- Conclusion

Therefore, the difference quotient simplifies to: \[ \frac{1}{\sqrt[3]{x+h}^2 + \sqrt[3]{x+h} \cdot \sqrt[3]{x} + \sqrt[3]{x}^2} \]

Key concepts

factoring formula

The factoring formula is a powerful tool in algebra that helps simplify complex expressions. For our problem, knowing how to factor expressions using the formula for the difference of cubes, \[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\], is crucial.
In this specific case, we set \(a = \sqrt[3]{x + h}\) and \(b = \sqrt[3]{x}\.\)
This means \(a^3 = x + h\) and \(b^3 = x\.\)
So, the expression \( a^3 - b^3 \) becomes \( (x + h) - x = h\), allowing our difference quotient to be factored neatly, making it easier to solve.

cube roots

Cube roots are the values that, when multiplied by themselves three times, yield the original number.
For instance, \( \sqrt[3]{8} = 2\) because \( 2 \times 2 \times 2 = 8 \).
In the function \(f(x) = \sqrt[3]{x}\), you deal with the cube roots of variables. This exercise utilizes cube roots to simplify the difference quotient, ultimately making algebraic manipulation more straightforward once you recognize how cube roots play into the factoring formula mentioned earlier.

simplifying expressions

Simplifying expressions transforms them into a more manageable and often more insightful form.
In simplifying our difference quotient, we follow these steps:

• First, write down the formula for the difference quotient and substitute the function \(f(x) = \sqrt[3]{x}\).
• Second, recognize the structure that matches the factoring formula.
• Third, rewrite the expression in its factored form using the hint provided.
• Finally, cancel common terms and simplify the leftover expressions.
  With each simplification step, our expression becomes clearer, making the solution process easier to follow.

algebraic manipulation

Algebraic manipulation refers to the process of using algebraic methods to rearrange and simplify expressions or equations.
In our example, applying the hint involves several key algebraic techniques:

• Substitution: Replace \(a\) and \(b\) with \(\sqrt[3]{x + h}\) and \(\sqrt[3]{x}\).
• Factoring: Apply the difference of cubes formula.
• Division: Simplify \( \frac{h}{h} = 1\).
• Simplification: Break down the expression to its simplest form.
  Throughout the steps, each technique helps in making the final quotient more understandable, reducing the complexity at each stage until we reach the solution.