Problem 8 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \sqrt[3]{\frac{x}{8}} $$

Step-by-Step Solution

Answer

\( \frac{\sqrt[3]{x}}{2} \)

Step 1: Understand the expression

The given expression is \( \sqrt[3]{\frac{x}{8}} \). This is the cube root of the fraction \( \frac{x}{8} \). Our aim is to simplify it.

Step 2: Apply the property of cube roots to fractions

Recall that the cube root of a quotient \( \sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}} \). We will use this property to separate the numerator and the denominator in the cube root expression.

Step 3: Simplify the cube root of the numerator

The numerator is \( x \), so we take the cube root of \( x \), which is \( \sqrt[3]{x} \) or simply \( x^{1/3} \).

Step 4: Simplify the cube root of the denominator

The denominator is 8. The cube root of 8 is 2 because \( 2^3 = 8 \). So, \( \sqrt[3]{8} = 2 \).

Step 5: Combine the results

Combine the simplified results from the numerator and the denominator: \( \frac{\sqrt[3]{x}}{2} \). This is the simplified form of the original expression.

Key Concepts

Simplifying Expressions

Simplifying expressions means breaking down mathematical phrases into their simplest form. Here, we begin by understanding what needs to be simplified. The expression, \( \sqrt[3]{\frac{x}{8}} \), involves finding the cube root of a fraction.
Remember that cube roots are about finding a number that, when multiplied by itself three times, gives the original number. In the case of fractions, we apply this process to both the numerator (the top part of the fraction) and the denominator (the bottom part).

• For expressions with fractions under a cube root, separate the numerator and the denominator first.
• Simplify each part individually before bringing them together again.

So, for \( \sqrt[3]{\frac{x}{8}} \), simplify to \( \frac{\sqrt[3]{x}}{\sqrt[3]{8}} \). Replacing \( \sqrt[3]{8} \) with 2 (because \( 2^3 = 8 \)), gives the simplified expression \( \frac{\sqrt[3]{x}}{2} \).

Fractional Exponents

Fractional exponents offer an alternative notation for roots, helping to simplify computations. When we see \( x^{1/3} \), it signifies the cube root of \( x \).
The whole idea is that fractional exponents turn root expressions into something more manageable in algebraic contexts.

• The "fraction" part of the exponent indicates the type of root: "3" in \( x^{1/3} \) relates to the cube root.
• This concept allows operations like multiplication and division to be done straightforwardly with exponents.

Applying this idea to \( \sqrt[3]{x} \), we can write it as \( x^{1/3} \). The power of using fractional exponents is it turns complex root operations into simple exponent arithmetic.

Properties of Exponents

Understanding the properties of exponents is crucial for simplifying expressions and solving complex problems. These properties provide the rules we follow when working with powers. Some fundamental properties include:

• \( x^{a+b} = x^a \cdot x^b \) : Add exponents when multiplying like bases.
• \( (x^a)^b = x^{a\cdot b} \) : Multiply exponents when raising a power to another power.
• \( x^{-a} = \frac{1}{x^a} \) : Convert negative exponents into fractions.
• \( x^{0} = 1 \) : Any non-zero base raised to the power of zero equals one.

Applying these properties helps simplify expressions easily. For example, knowing \( (x^{1/3})^3 = x \), enables more intuitive solutions to root problems. Learning and practicing these rules is key to mastering algebra and makes handling expressions like \( \sqrt[3]{\frac{x}{8}} \) straightforward.