Problem 44 - College Algebra and Calculus: An Applied Approach

Question

Rewrite the expression by rationalizing the denominator. Simplify your answer.\(\frac{5}{\sqrt[3]{(5 x)^{2}}}\)

Step-by-Step Solution

Answer

The simplified, rationalized form of the original expression \(\frac{5}{\sqrt[3]{(5x)^{2}}}\) is \(x^{-2/3}*5^{1/3}\)

Step 1: Write down the expression

The given expression is \(\frac{5}{\sqrt[3]{(5 x)^{2}}}\)

Step 2: Rewrite the expression

The numerator remains the same. The denominator, which is a cube root, can be written as \((5x)^{2/3}\). Hence the expression can be rewritten as \(\frac{5}{(5x)^{2/3}}\)

Step 3: Rationalize the denominator

To get rid of \((5x)^{2/3}\) in the denominator, multiply both the numerator and denominator by \((5x)^{1/3}\). This will give: \(\frac{5*(5x)^{1/3}}{(5x)^{2/3}*(5x)^{1/3}} = \frac{5*(5x)^{1/3}}{(5x)}\)

Step 4: Simplify the expression

The \(x\) in the denominator can cancel out one of the \(x\) in the numerator, and likewise for the 5 in the numerator and denominator. Therefore the expression simplifies to: \(\frac{5^{1/3}}{x^{2/3}}\) or \(x^{-2/3}*5^{1/3}\)

Key Concepts

Cube Roots

Cube roots play an essential role in solving algebraic expressions involving cube roots in the denominator.
When we have a cube root, it means finding a number which, when multiplied by itself three times, gives the original number.
For example, the cube root of 8 is 2 because when 2 is multiplied by itself three times (2 * 2 * 2), the result is 8.

In expressions such as \(\frac{5}{\sqrt[3]{(5x)^2}}\), the cube root affects how we manipulate the expression, particularly in the context of rationalizing the denominator.
This is because cube roots in the denominator can make calculations more complex, so getting rid of them often simplifies the problem.

• Conversion from cube root: \(\sqrt[3]{(5x)^2} = (5x)^{2/3}\)
• Determining factors that can simplify the expression
• Observing patterns to rationalize the denominator

Algebraic Expressions

Algebraic expressions consist of numbers and variables combined using mathematical operations.
In this context, the expression \(\frac{5}{(5x)^{2/3}}\) involves a constant (the number 5) and a variable with exponentiation involving cube roots.

Understanding algebraic expressions is crucial as it allows us to apply rules and properties to simplify, manipulate, and solve them efficiently.
These expressions can range from simple to very complex, depending on how many terms and operations they involve.

• Variables represent unknown values, which can change
• Coefficients are the numbers multiplying the variables
• Exponents show the power to which a term is raised

Familiarity with these helps in rationalizing the denominator through multiplication techniques and applying exponent rules like when multiplying powers with the same base.

Simplification

Simplification is the process of reducing an expression to its simplest form.
It often involves rationalizing denominators, canceling terms, and combining like terms.
For the expression \(\frac{5(5x)^{1/3}}{5x}\), simplification includes:

• Cancelling common factors; in this case, the 5 and x from the numerator and denominator
• Applying exponent rules to simplify powers of the same base

By conducting these steps, we arrive at a much more manageable expression like \[x^{-2/3}5^{1/3}\], which is simplified with rational components.

This process not only makes calculations easier but also helps in grasping the fundamental structures of the algebraic expression.
Mastery in simplification ensures the ability to approach more complex algebraic problems with ease.