Problem 79 - Elementary and Intermediate Algebra

Question

Rationalize each denominator. All variables represent positive real numbers. $$ \sqrt[3]{\frac{5}{16}} $$

Step-by-Step Solution

Answer

The rationalized form is \( \frac{\sqrt[3]{20}}{4} \).

Step 1: Understand the Expression

The expression given is \( \sqrt[3]{\frac{5}{16}} \). This is a cube root of a fraction, \( \frac{5}{16} \), where \( 5 \) is the numerator and \( 16 \) is the denominator.

Step 2: Rationalize the Denominator

To rationalize the expression \( \sqrt[3]{\frac{5}{16}} \), consider that the denominator \( 16 \) should be a perfect cube. Currently, \( 16 = 2^4 \), which is not a complete cube. Find the smallest number that, when multiplied by \( 16 \), results in a perfect cube.

Step 3: Determine the Factor to Multiply

Since \( 16 = 2^4 \), we need to multiply by \( 2^2 \) (i.e., \( 4 \)) to get a perfect cube, since \( 2^4 \times 2^2 = 2^6 = (2^2)^3 = 64 \), which is \( (4)^3 \).

Step 4: Multiply the Numerator and Denominator

Multiply the fraction by \( \frac{4}{4} \) to keep it equivalent. Hence, \( \sqrt[3]{\frac{5}{16}} \times \frac{4}{4} = \sqrt[3]{\frac{20}{64}} \).

Step 5: Simplify the Expression

Now that the denominator is a perfect cube, simplify both the numerator and the denominator. The expression is now \( \sqrt[3]{\frac{20}{64}} = \frac{\sqrt[3]{20}}{\sqrt[3]{64}} \). Since \( \sqrt[3]{64} = 4 \), the expression becomes \( \frac{\sqrt[3]{20}}{4} \).

Step 6: Final Simplification

Since 20 cannot be further simplified into a smaller cube root, the expression \( \frac{\sqrt[3]{20}}{4} \) is the rationalized form of the original expression.

Key Concepts

Cube Roots

Cube roots help us find a number that, when used three times in a multiplication, results in the given value. Consider the cube root like this: what number cubed (multiplied by itself twice more) gives us the value we have?
In our exercise, we're dealing with the cube root of a fraction, specifically \( \sqrt[3]{\frac{5}{16}} \).
The symbol \( \sqrt[3]{} \) means we need a number that multiplied by itself twice (i.e., cubed) gives the number beneath the symbol.
For example, if you have \( \sqrt[3]{8} = 2 \), it's because \( 2 \times 2 \times 2 = 8 \).

• Cube roots apply to both whole numbers and fractions.
• When extracting cube roots from fractions, we focus independently on the numerator and the denominator.

When rationalizing the denominator of a cube root, our goal is to transform the fraction involving a cube root into one that has a neat, perfect cube in the denominator.

Perfect Cubes

Perfect cubes are numbers that can be expressed as the cube of an integer. Just like square numbers, which are the squares of integers, perfect cubes result from multiplying an integer by itself two additional times.
For example, some of the smallest perfect cubes include 1, 8, and 27, because they result from \(1^3\), \(2^3\), and \(3^3\) respectively.
In our exercise, when we started with the fraction \( \frac{5}{16} \), the denominator \( 16 \) was not a perfect cube, since there isn't a whole number \( n \) such that \( n^3 = 16 \).
To make \( 16 \) into a perfect cube, we calculated what it would take to scale up:

• We realized \( 16 = 2^4 \), which isn't a cube.
• By multiplying \( 16 \) by \( 4 \) (because \( 4 = 2^2 \)), we could create \( 64 \), which is a perfect cube because \( 4^3 = 64 \).

This method of scaling up helps achieve a rationalized denominator that simplifies further handling of expressions.

Simplifying Expressions

Simplifying expressions can make complex math easier to manage. In mathematical expression simplification, we aim to make the expression as easy as possible to understand and work with, especially when cube roots are involved.
When dealing with complex expressions, simplifying helps us focus on the core parts without distractions from complicating factors.

• With \( \sqrt[3]{\frac{20}{64}} \), it converts to \( \frac{\sqrt[3]{20}}{\sqrt[3]{64}} \).
• Since \( \sqrt[3]{64} = 4 \), the denominator becomes simpler.
• We reach the simplified expression as \( \frac{\sqrt[3]{20}}{4} \).

Here, we focus on operations that either do not change the expression's value or make it simpler. Reducing the fraction, or simplifying roots helps to present the expression in a much simpler structure, which is easier to interpret or to calculate in further steps.