Problem 80 - Intermediate Algebra

Question

Rationalize each denominator. All variables represent positive real numbers. $$ \sqrt[3]{\frac{2}{81}} $$

Step-by-Step Solution

Answer

\( \frac{2}{9} \cdot \sqrt[3]{6} \) is the rationalized form.

Step 1: Identify the Problem

We need to rationalize the denominator of the expression \( \sqrt[3]{\frac{2}{81}} \). The goal is to remove the cube root from the denominator to make it a rational number.

Step 2: Rewrite the Cube Root

Express the cube root in terms of exponents: \( \sqrt[3]{\frac{2}{81}} = \left(\frac{2}{81}\right)^{\frac{1}{3}} \). This means we are taking the cube root of both the numerator and the denominator.

Step 3: Rationalize the Denominator

To rationalize \( 81^{\frac{1}{3}} \), we multiply by \( 81^{\frac{2}{3}} \) because: \((81^{\frac{1}{3}}) \times (81^{\frac{2}{3}}) = 81^{\frac{3}{3}} = 81^1 = 81\). Multiply numerator and denominator by \( 81^{\frac{2}{3}} \): \[ \frac{2^{\frac{1}{3}} \cdot 81^{\frac{2}{3}}}{81^{\frac{3}{3}}} = \frac{2^{\frac{1}{3}} \cdot 81^{\frac{2}{3}}}{81} \]

Step 4: Simplify the Expression

Calculate \( 81^{\frac{2}{3}} \): Recall that 81 is \( 3^4 \), so \( 81^{\frac{2}{3}} = (3^4)^{\frac{2}{3}} = 3^{\frac{8}{3}} \). This can be split into \( 3^{\frac{8}{3}} = 3^{2} \times 3^{\frac{2}{3}} = 9 \times 3^{\frac{2}{3}} \).Thus, the expression simplifies to:\[ \frac{2^{\frac{1}{3}} \cdot 9 \cdot 3^{\frac{2}{3}}}{81} \]

Step 5: Combine and Simplify

Now, simplify the entire expression:\[ \frac{18 \cdot (2 \cdot 3^2)^{\frac{1}{3}}}{81} \]This gives us:\[ \frac{18 \cdot 6^{\frac{1}{3}}}{81} = \frac{18 \cdot 6^{\frac{1}{3}}}{81} = \frac{18}{81} \cdot 6^{\frac{1}{3}} \]Simplify \( \frac{18}{81} \) to get \( \frac{2}{9} \cdot 6^{\frac{1}{3}} \).

Step 6: Final Rationalized Form

The rationalized form of the original expression is:\[ \frac{2}{9} \cdot \sqrt[3]{6} \]

Key Concepts

Cube Roots

Cube roots refer to an operation that determines a number which, when multiplied by itself twice, yields the original number. If you have a number \( x \), its cube root is represented as \( \sqrt[3]{x} \) or equivalently as \( x^{\frac{1}{3}} \). Cube roots are particularly useful in problems where you want to reverse or simplify a cubed expression. For example, if \( 2^3 = 8 \), then \( \sqrt[3]{8} = 2 \). This allows us to simplify expressions and solve equations by reducing higher powers into more manageable forms.

When rationalizing a denominator containing a cube root, like in the problem \( \sqrt[3]{\frac{2}{81}} \), the aim is to eliminate the radical and make the denominator a rational number. We achieve this by multiplying the denominator by another expression that brings the entire term to an integer power, specifically using cube root properties effectively. Ultimately, this involves familiarity with cube roots to transform and simplify each term efficiently.

Exponents

Exponents, often referred to as powers, are a way of expressing repeated multiplication of a number by itself. For instance, \( 3^4 \) means that 3 is multiplied by itself four times, i.e., \( 3 \times 3 \times 3 \times 3 \). For cube roots, an exponent of \( \frac{1}{3} \) signifies a cube root itself, for example, \( x^{\frac{1}{3}} \) is equivalent to \( \sqrt[3]{x} \).

Using exponent rules can greatly simplify rationalizing expressions. Multiplying powers with the same base involves adding the exponents: \( a^m \times a^n = a^{m+n} \). Similarly, raising a power to another power means multiplying exponents: \( (a^m)^n = a^{m \times n} \). These rules help to break down complex expressions, making them easier to manipulate and interpret.

• Example: \((81^{\frac{1}{3}}) \times (81^{\frac{2}{3}}) = 81^{\frac{3}{3}} = 81^1 = 81\)

Knowing how to effectively apply these properties is essential for students tackling problems in mathematics and its applications.

Simplifying Expressions

Simplifying expressions is a core algebraic skill which involves reducing expressions to their most basic and concise form. This process typically involves using various algebraic techniques such as distributing, combining like terms, and reducing fractions. Consider the given problem, where we need to simplify \( \frac{2^{\frac{1}{3}} \cdot 9 \cdot 3^{\frac{2}{3}}}{81} \).

Simplifying involves:

• Breaking down: Knowing that \( 81 = 9 \times 9 \), simplifies to \( 81 = 3^4 \).
• Combining like terms: Use properties of exponents to combine similar bases like \( 3^{2/3} \) and other terms.
• Reducing: Fractions like \( \frac{18}{81} \) can be simplified to \( \frac{2}{9} \).

Mastering these techniques aids in rationalizing denominators, leading to clearer and more easily understood expressions. This simplification not only makes it easier to carry out further calculations but also helps in verifying results more effectively.