Problem 56 - Algebra and Trigonometry

Question

Simplify the expression and eliminate any negative exponents \((\mathrm{s}) .\) Assume that all letters denote positive numbers. \(\left(\frac{-2 x^{1 / 3}}{y^{1 / 2} z^{1 / 6}}\right)^{4}\)

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{16 x^{4/3}}{y^2 z^{2/3}} \).

Step 1: Expand the Expression

The expression is \( \left( \frac{-2 x^{1/3}}{y^{1/2} z^{1/6}} \right)^4 \). Applying the power property \( (a/b)^n = a^n / b^n \), expand the expression: \( \frac{(-2)^4 (x^{1/3})^4}{(y^{1/2})^4 (z^{1/6})^4} \).

Step 2: Simplify the Numerator

The numerator is \((-2)^4 (x^{1/3})^4\). Simplify \((-2)^4 = 16\) and \((x^{1/3})^4 = x^{4/3}\). Therefore, the numerator becomes \(16 x^{4/3}\).

Step 3: Simplify the Denominator

For the denominator, calculate \((y^{1/2})^4 = y^2\) and \((z^{1/6})^4 = z^{2/3}\). Thus, the denominator becomes \(y^2 z^{2/3}\).

Step 4: Combine Simplified Components

Combine the simplified numerator and denominator: \( \frac{16 x^{4/3}}{y^2 z^{2/3}} \). This expression has no negative exponents and is simplified.

Key Concepts

Exponent Rules

Exponent rules are crucial for simplifying expressions, especially those involving powers. The core principles include:

• Product of Powers Rule: Multiply exponents if the bases are identical (e.g., \(a^m \times a^n = a^{m+n}\)).
• Power of a Power Rule: Multiply exponents (e.g., \((a^m)^n = a^{m \cdot n}\)).
• Quotient of Powers Rule: Subtract exponents if the bases are identical (e.g., \(\frac{a^m}{a^n} = a^{m-n}\)).
• Negative Exponent Rule: Convert negative exponents to positive by flipping their base (e.g., \(a^{-m} = \frac{1}{a^m}\)).

In the original exercise, these rules are used to expand and then simplify the expression. Knowing these rules helps in handling complex expressions without confusion.

Rational Exponents

Rational exponents are used to express roots and powers in a single term. They follow the format \(a^{\frac{m}{n}}\), where \(m\) is the power and \(n\) is the root. For example, \(x^{1/2}\) indicates the square root of \(x\) while \(x^{1/3}\) indicates the cube root.

Rational exponents allow you to simplify expressions that involve both roots and higher powers. In the problem, using \(x^{1/3}\) and simplifying it to a higher power, as in \((x^{1/3})^4 = x^{4/3}\), can transform expressions into a more manageable form.

• Simplifies notation by combining roots and powers.
• Makes it easier to apply exponent rules.

Power Property

The power property helps when raising powers to others. It states that \((a^m)^n = a^{m\cdot n}\). This is particularly useful when dealing with expressions where both the base and the exponent have exponents.

In the solving steps, the power property is applied to transform components like \((x^{1/3})^4\) into \(x^{4/3}\). With this property, you can reduce complex layered expressions into simpler, more readable ones.

• Rationalizes complexity by simplifying step by step.
• Used in both expanding and simplifying stages.

Grasping this rule is key to developing a solid understanding of algebraic manipulation.

Positive Numbers Assumption

Assuming that variables represent positive numbers is crucial in simplifying expressions with rational exponents. This assumption avoids complications that arise with negative numbers, especially with even roots.

For instance, the expression \(y^{1/2}\) represents \(\sqrt{y}\). If \(y\) were negative, taking the square root wouldn't yield a real number.

• Ensure expressions remain mathematically valid and simple.
• Avoids the complexity of imaginary numbers.

This assumption supports the simplification of expressions while keeping solutions valid and straightforward.