Problem 64 - Elementary and Intermediate Algebra

Question

Rationalize each denominator. $$ \frac{2}{\sqrt[3]{a}} $$

Step-by-Step Solution

Answer

The rationalized expression is \( \frac{2 \sqrt[3]{a^2}}{a} \).

Step 1: Identify the Problem

The problem asks to rationalize the denominator of the expression \( \frac{2}{\sqrt[3]{a}} \). Currently, the denominator is not a rational number because it contains a cube root.

Step 2: Multiply by the Conjugate Fraction

To rationalize the denominator which is \( \sqrt[3]{a} \), we multiply both the numerator and the denominator by \( \sqrt[3]{a^2} \). This is because \( \sqrt[3]{a} \times \sqrt[3]{a^2} = \sqrt[3]{a^3} = a \), which is a rational number.

Step 3: Apply the Multiplication

We perform the multiplication, resulting in: \[\frac{2 \times \sqrt[3]{a^2}}{\sqrt[3]{a} \times \sqrt[3]{a^2}}\]Simplifying the denominator, we obtain:\[\frac{2 \sqrt[3]{a^2}}{a}\]

Step 4: Simplify the Expression

The expression \( \frac{2 \sqrt[3]{a^2}}{a} \) is simplified such that the denominator is now a rational number \( a \), and the entire expression is simplified as much as possible while keeping the denominator rationalized.

Key Concepts

Cube Roots

The cube root of a number is a special value that, when multiplied by itself three times, gives the original number. It is symbolized as \( \sqrt[3]{x} \). For example, \( \sqrt[3]{8} = 2 \) because \( 2 \times 2 \times 2 = 8 \). In our original problem, we have \( \sqrt[3]{a} \), which is the cube root of \( a \). This tells us:

• If you raise this cube root to the power of three, you'll get the original number \( a \).
• Cube roots help in simplifying expressions and are vital when working with cube-shaped values in geometry or for finding dimensions.
• Unlike square roots, every real number has a real cube root, which can be positive or negative.

These properties are used strategically, especially when rationalizing denominators.

Rational Numbers

In mathematics, a rational number is any number that can be expressed as the quotient or fraction \( \frac{p}{q} \) of two integers, with the numerator \( p \) and the denominator \( q \), where \( q eq 0 \). Rational numbers include integers, fractions, and finite or repeating decimals.

• They are called 'rational' since they can be written as a ratio.
• The number \( a \) in the denominator, after rationalizing \( \sqrt[3]{a} \), is a rational number because it can be expressed as \( \frac{a}{1} \).
• Converting complex roots in the denominator into rational numbers simplifies expressions, making equations easier to work with.

Understanding how to convert numbers into rational form is crucial for performing arithmetic operations and algebraic manipulation efficiently.

Algebraic Expressions

Algebraic expressions combine numbers, variables, and arithmetic operations to create a new value or equation. An expression like \( \frac{2}{\sqrt[3]{a}} \) contains both a number (2) and a variable part (\( \sqrt[3]{a} \)). This highlights a few important aspects:

• Variables represent numbers and allow expressions to describe a set of numbers rather than a single value.
• Simplifying algebraic expressions often involves rationalizing denominators, which means removing irrational numbers from the bottom of a fraction.
• In our exercise, multiplying by \( \sqrt[3]{a^2} \) helps to transform \( \sqrt[3]{a} \) into \( a \), achieving a neat algebraically simplified expression \( \frac{2 \sqrt[3]{a^2}}{a} \).

Algebraic expressions provide a framework for solving complex mathematical problems, expressing real-world relationships mathematically, and communicating mathematical ideas efficiently.