Problem 41 - Intermediate Algebra

Question

Add or subtract. $$ \sqrt[3]{\frac{16}{27}}-\frac{\sqrt[3]{54}}{6} $$

Step-by-Step Solution

Answer

\( \frac{1}{6}\sqrt[3]{2} \)

Step 1: Simplify the Expression Inside the Cube Root

The expression given is \( \sqrt[3]{\frac{16}{27}} - \frac{\sqrt[3]{54}}{6} \). First, simplify \( \sqrt[3]{54} \) as follows. Since 54 can be factored into \( 27 \times 2 \), we have \( \sqrt[3]{54} = \sqrt[3]{27 \times 2} = \sqrt[3]{27} \times \sqrt[3]{2} = 3 \times \sqrt[3]{2} \).

Step 2: Simplify \( \frac{\sqrt[3]{54}}{6} \)

Substitute \( \sqrt[3]{54} = 3\sqrt[3]{2} \) from Step 1 into \( \frac{\sqrt[3]{54}}{6} \). This gives \( \frac{3\sqrt[3]{2}}{6} = \frac{1}{2}\sqrt[3]{2} \).

Step 3: Simplify \( \sqrt[3]{\frac{16}{27}} \)

Now, simplify \( \sqrt[3]{\frac{16}{27}} \). Since 16 is \( 2^4 \) and 27 is \( 3^3 \), \( \sqrt[3]{\frac{16}{27}} = \frac{\sqrt[3]{16}}{\sqrt[3]{27}} = \frac{\sqrt[3]{2^4}}{\sqrt[3]{3^3}} = \frac{2^{4/3}}{3} \). This simplifies to \( \frac{2\cdot 2^{1/3}}{3} = \frac{2}{3}\sqrt[3]{2} \).

Step 4: Perform the Subtraction

Now subtract \( \frac{\sqrt[3]{54}}{6} = \frac{1}{2}\sqrt[3]{2} \) from \( \sqrt[3]{\frac{16}{27}} = \frac{2}{3}\sqrt[3]{2} \). This gives \( \frac{2}{3}\sqrt[3]{2} - \frac{1}{2}\sqrt[3]{2} \).

Step 5: Subtract the Like Terms

Subtract \( \frac{1}{2} \) from \( \frac{2}{3} \) using a common denominator, which is 6. So we rewrite \( \frac{2}{3}\sqrt[3]{2} \) as \( \frac{4}{6}\sqrt[3]{2} \) and \( \frac{1}{2}\sqrt[3]{2} \) as \( \frac{3}{6}\sqrt[3]{2} \). The subtraction is \( \frac{4}{6}\sqrt[3]{2} - \frac{3}{6}\sqrt[3]{2} = \frac{1}{6}\sqrt[3]{2} \).

Step 6: Final Result

The simplified result of the original expression is \( \frac{1}{6}\sqrt[3]{2} \).

Key Concepts

Simplifying Radicals

When working with cube roots, or radicals, a key step is to simplify them to their most basic form. This often requires factoring numbers under the radical sign to break them into more manageable parts. For instance, with the term \( \sqrt[3]{54} \), you can express it as \( \sqrt[3]{27 \times 2} \). This simplifies further into \( \sqrt[3]{27} \times \sqrt[3]{2} \). Since 27 is a perfect cube, equal to \( 3^3 \), you can take the cube root of 27 out as 3, simplifying \( \sqrt[3]{54} \) to \( 3 \times \sqrt[3]{2} \). By understanding the properties of exponents and roots, you can make complex radicals simpler to deal with.
Always look for perfect cubes when simplifying cube roots to streamline your calculations.

Arithmetic Operations

Arithmetic operations, such as addition and subtraction, are crucial when handling expressions involving radicals. It's essential to deal with like terms, just as you would when combining polynomial expressions. Like terms have the same radical part. In the given example, subtracting \( \frac{1}{2}\sqrt[3]{2} \) from \( \frac{2}{3}\sqrt[3]{2} \), you focus on the coefficients of the like radical \( \sqrt[3]{2} \). To perform this subtraction accurately, find a common denominator.
In the example, the common denominator between 2 and 3 is 6, allowing the fractions \( \frac{2}{3} \) and \( \frac{1}{2} \) to be rewritten as \( \frac{4}{6} \) and \( \frac{3}{6} \). After aligning these denominators, subtract \( \frac{3}{6}\sqrt[3]{2} \) from \( \frac{4}{6}\sqrt[3]{2} \) to get \( \frac{1}{6}\sqrt[3]{2} \).
This step highlights the importance of organizing your work and systematically simplifying fractional coefficients before proceeding with the operation.

Rational Expressions

Rational expressions involve fractions that contain radicals either in the numerator, the denominator, or both. When dealing with these expressions, it's imperative to seek simplification opportunities. Take \( \sqrt[3]{\frac{16}{27}} \) as demonstrated. This can be expressed separately as \( \frac{\sqrt[3]{16}}{\sqrt[3]{27}} \). Recognizing that 16 is \( 2^4 \) and 27 is \( 3^3 \), you can simplify each component.
The cube root of 16 yields \( 2^{4/3} \), which can be further simplified to \( 2 \times 2^{1/3} \). The cube root of 27 is simply 3. Thus, the simplified version of \( \sqrt[3]{\frac{16}{27}} \) is \( \frac{2}{3}\sqrt[3]{2} \).
By fractionating radicals and manipulating the terms with basic root rules, you manage to simplify the expression into a more comprehensible form, reducing complexity and making further arithmetic operations easier to perform.