Problem 19 - Algebra A Combined Function

Question

Rationalize each denominator. Assume that all variables represent positive real numbers. \(\sqrt[3]{\frac{3}{5}}\)

Step-by-Step Solution

Answer

\(\frac{\sqrt[3]{75}}{5}\) is the rationalized form.

Step 1: Identify the Rationalization Requirement

The problem requires us to rationalize the denominator of \(\sqrt[3]{\frac{3}{5}}\). This means we need to eliminate the cube root in the denominator to make it a rational number.

Step 2: Determine the Rationalizing Factor

To rationalize \(\sqrt[3]{5}\), we need to multiply the numerator and the denominator by a suitable term. In this case, the suitable term is \(\sqrt[3]{25}\), as \((\sqrt[3]{5}) \times (\sqrt[3]{25}) = 5\).

Step 3: Multiply to Rationalize

Multiply both the numerator and the denominator by the rationalizing factor:\[ \frac{\sqrt[3]{3} \times \sqrt[3]{25}}{\sqrt[3]{5} \times \sqrt[3]{25}} = \frac{\sqrt[3]{75}}{5}.\]

Step 4: Simplify the Expression

The numerator can be written as \(\sqrt[3]{75}\), which does not simplify further since \(75 = 3 \times 5^2\). Therefore, the rationalized form of the expression is \(\frac{\sqrt[3]{75}}{5}\).

Key Concepts

Understanding Cube Roots

When we talk about cube roots, we're looking at the number that, when multiplied by itself twice, gives the original value. For example, the cube root of 8 is 2 because \(2 \times 2 \times 2 = 8\). Cube roots are useful when dealing with volumes and other three-dimensional calculations. They are denoted using the radical symbol with a small 3 above it: \(\sqrt[3]{...}\).
Cube roots are not as straightforward as square roots, since they involve three repetitions of a number. However, understanding them becomes easier with practice.

• If \(a^3 = b\), then \(\sqrt[3]{b} = a\).
• Cube roots can be applied to both positive and negative numbers, unlike square roots.
• Finding a cube root involves recognizing cubes and potentially simplifying the expression.

Exploring Rational Numbers

Rational numbers are numbers that can be expressed as the fraction of two integers. They can be written in the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers, and \(b\) is not zero. This concept includes ordinary numbers like 1, 2, or 3, but also decimals that repeat or terminate, such as 0.5 or 0.333....
Rational numbers are crucial in mathematics because they form the basic building blocks of arithmetic and algebra. They allow us to perform a wide range of operations and are very handy for solving equations. Here are some characteristics of rational numbers:

• They can be positive or negative.
• Every integer is a rational number since it can be expressed as itself divided by 1.
• If a number has a repeating or terminating decimal expansion, it is rational.

Understanding rational numbers provides a foundation for more complex mathematical concepts such as rationalization.

The Role of Denominators in Rationalization

Denominators represent the division part of a fraction, showing how many parts the whole is divided into. In rationalization, particularly with radicals, we aim to convert any irrational part of the denominator into a rational form. This simplifies calculations and comparisons between fractions.
Rationalization of the denominator, as shown in the exercise \(\sqrt[3]{\frac{3}{5}}\), involves eliminating radicals or cube roots from the denominator to make the entire value fractionally rational.

• By multiplying both the numerator and the denominator by an appropriate term, you ensure rationality in the denominator.
• In our example, multiplying by \(\sqrt[3]{25}\) helps transform \(\sqrt[3]{5}\) into 5, a rational number.
• Rationalizing makes expressions easier to handle and often clearer in values and operations.

Overall, understanding denominators' role in rationalization helps simplify expressions, making them more intuitive and manageable.