Problem 76 - Algebra and Trigonometry

Question

\(69-82\) . Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers. $$ \frac{\sqrt[3]{8 x^{2}}}{\sqrt{x}} $$

Step-by-Step Solution

Answer

The simplified form is \( 2x^{1/6} \).

Step 1: Express Radicals with Rational Exponents

The expression \( \sqrt[3]{8x^2} \) can be rewritten using rational exponents as \( (8x^2)^{1/3} \). Similarly, the expression \( \sqrt{x} \) can be rewritten as \( x^{1/2} \). Thus, our expression becomes \( \frac{(8x^2)^{1/3}}{x^{1/2}} \).

Step 2: Simplify the Numerator

In the expression \( (8x^2)^{1/3} \), the constant 8 can be rewritten as \( 2^3 \). Therefore, \( (8x^2)^{1/3} = (2^3x^2)^{1/3} = 2x^{2/3} \).

Step 3: Divide Using the Laws of Exponents

Using the laws of exponents, we divide the expression \( \frac{2x^{2/3}}{x^{1/2}} \). This simplifies to \( 2x^{2/3 - 1/2} \). Subtract the exponents: \( \frac{2}{3} - \frac{1}{2} = \frac{4}{6} - \frac{3}{6} = \frac{1}{6} \).

Step 4: Write the Final Expression

After simplifying the exponents, the expression is \( 2x^{1/6} \). This is the simplified form using rational exponents.

Key Concepts

Laws of Exponents

The laws of exponents are powerful rules that help us manipulate expressions involving powers. They provide a consistent method for simplifying complex expressions. Here are a few important rules:

• Product of Powers: When multiplying like bases, add the exponents: \( a^m \times a^n = a^{m+n} \).
• Quotient of Powers: When dividing like bases, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Power of a Power: To raise a power to another power, multiply the exponents: \( (a^m)^n = a^{m\cdot n} \).
• Negative Exponent: A negative exponent represents a reciprocal: \( a^{-n} = \frac{1}{a^n} \).
• Zero Exponent: Any non-zero base raised to the zero power equals one: \( a^0 = 1 \).

In our original problem, we used the quotient of powers law to simplify the expression. By applying this rule, we easily managed to subtract the exponents of x in the expression \( \frac{x^{2/3}}{x^{1/2}} \). This resulted in the simplified form \( x^{1/6} \). Understanding these rules makes the process of dealing with powers more efficient and manageable.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form while maintaining the value. It allows us to express complex ideas in a straightforward manner.

• First, convert all roots and radicals into rational exponents if necessary. This means turning expressions like \( \sqrt{a} \) into \( a^{1/2} \) and \( \sqrt[3]{a} \) into \( a^{1/3} \).
• Next, apply the laws of exponents to perform operations like multiplication, division, or exponentiation.
• Simplify coefficients separately from variables.
• If possible, combine like terms for further simplification.

In the provided problem, the expression \( \frac{\sqrt[3]{8x^2}}{\sqrt{x}} \) was simplified by first converting the radicals into rational exponents, leading to \( \frac{(8x^2)^{1/3}}{x^{1/2}} \). Then, simplifying the coefficients and applying the appropriate laws of exponents enabled us to arrive at \( 2x^{1/6} \). This systematic approach helps in reducing potential errors.

Rational Expression

Rational expressions are quotients involving polynomials. They act similarly to rational numbers, but instead of integers, they involve variables and exponents. Understanding how to work with rational expressions is crucial because they appear frequently in algebra.

• Definition: A rational expression takes the form \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \).
• Operations: You can add, subtract, multiply, or divide them, similar to numeric fractions. However, ensuring the denominators are not zero is essential.
• Simplification: Simplifying involves reducing the expression to its lowest terms by factoring and canceling common factors in the numerator and denominator.

In our case, we dealt with a rational expression of the form \( \frac{(8x^2)^{1/3}}{x^{1/2}} \). To simplify it, we employed rational exponents to ease division and applied the laws of exponents effectively. It is important to perform these operations meticulously to avoid errors, particularly when dealing with expressions that include roots and powers.