Problem 39 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-46: Simplify. $$ \left(8 x^{-2 / 3}\right) x^{1 / 6} $$

Step-by-Step Solution

Answer

The simplified expression is \( 8x^{-1/2} \).

Step 1: Distribute Exponents

First, look at the expression \( (8x^{-2/3})x^{1/6} \). Distribute the \( x \) terms so that you have \( 8x^{-2/3} \times x^{1/6} \).

Step 2: Apply Properties of Exponents

Use the properties of exponents to combine \( x^{-2/3} \) and \( x^{1/6} \). When multiplying terms with the same base, you add the exponents: \[ x^{-2/3 + 1/6} \].

Step 3: Find a Common Denominator

Find the common denominator for the fractions \( -2/3 \) and \( 1/6 \). The common denominator is 6. Rewrite the exponents as \(-4/6\) and \(1/6\).

Step 4: Add the Exponents

Add the exponents together: \(-4/6 + 1/6 = -3/6 = -1/2\).

Step 5: Simplify the Expression

Now substitute the new exponent back into the expression: \(8x^{-1/2}\). As the expression is already simplified, there is no need for further simplification.

Key Concepts

Properties of exponents

When working with algebraic expressions containing exponents, understanding the properties of exponents is essential. These properties provide a set of rules for simplifying expressions involving powers or indices. Here are some fundamental properties:

• Product of Powers Property: When multiplying two expressions with the same base, you add the exponents: \( a^m \times a^n = a^{m+n} \).
• Power of a Power Property: When taking a power of a power, you multiply the exponents: \( (a^m)^n = a^{m\times n} \).
• Power of a Product Property: When taking a power of a product, distribute the exponent to both factors: \( (ab)^n = a^n \times b^n \).

In the given exercise, you applied the "Product of Powers Property". You added the exponents of like bases \( x^{-2/3} \times x^{1/6} = x^{-2/3 + 1/6} \). These principles will help you manipulate and simplify expressions efficiently.

Rational exponents

Rational exponents are a way to represent roots of numbers using fractions as exponents. Understanding this concept involves understanding that a rational exponent \( \frac{m}{n} \) means the \(n\)-th root of a number raised to the \(m\)-th power: \( a^{m/n} = \sqrt[n]{a^m} \).

• Fractional exponents: The numerator is the power and the denominator is the root. For instance, \( x^{1/2} \) represents \( \sqrt{x} \).
• Negative exponents: A negative exponent signifies taking the reciprocal. \( x^{-m} = \frac{1}{x^m} \).

In the exercise, you dealt with fractional exponents like \(-2/3\) and \(1/6\). By finding a common denominator, you can add these fractions, simplifying them into a single rational exponent \( -1/2 \).

Combining like terms

Combining like terms is a crucial step in simplifying algebraic expressions. Like terms are terms that contain the same variables raised to the same power. Simplifying requires collecting these terms and combining them into a single term.

• The process involves adding or subtracting coefficients while keeping the variable part unchanged.
• For example, in an expression like \( 3x^2 + 5x^2 \), the like terms \(x^2\) are combined to give \(8x^2\).

In the provided problem, the focus was on the exponents. While the terms weren't combined in the traditional sense of adding and subtracting coefficients, understanding which terms are alike (having the base \( x \)) allowed for proper application of the exponent properties. This indirectly helped simplify the expression to its fullest extent through combining the effects of their exponents.