Problem 72 - Algebra 2

Question

To rationalize the denominator of \(\sqrt[3]{\frac{2}{9}},\) by what number would you multiply the numerator and denominator of the fraction? F. 2 G. 3 H. 6 J.9

Step-by-Step Solution

Answer

G. 3

Step 1: Analyze the fraction under the cube root

First, we need to recognize, that the fraction under the cube root is \(\frac{2}{9}\). This means, the cube root is applied to both the numerator (2) and the denominator (9).

Step 2: Implement the method to rationalize the denominator

To rationalize the denominator (get rid of the cube root in the denominator), the numerator and the denominator should be multiplied by the square of the number in the denominator, which is \(9^2\) or 81. However, the cubic root of 81 is not a whole number. Therefore, we need to find the cube that will eliminate the cubic root altogether, which means we need to multiply by the cube of 3 - being \(3^3\) or 27.

Step 3: Confirm the correct answer

The cube of 3 is 27, which is not an option in our answer choices. To derive the correct answer, take the cubic root of 27, which equals 3. Therefore, multiply both the numerator and denominator by 3, since the denominator already contained a 9, the product will be 27, which will rationalize the denominator.

Key Concepts

Cube Roots

Cube roots are an essential concept in mathematics and relate to the operation of "cubing," which involves raising a number to the power of three. The cube root of a number is another number which, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3, because 3 \(\times\) 3 \(\times\) 3 equals 27. This operation is denoted by the radical symbol with a small three, \(\sqrt[3]{}\), placed above the root.Understanding cube roots is important when dealing with values under cube radical expressions, as they are used for simplifying calculations or interpreting results.

• Cube Roots indicate that a number is multiplied by itself three times.
• The cube root of \(x^3\) is \(x\).
• Knowing how to find cube roots helps in simplifying complex algebraic expressions.

By mastering the concept of cube roots, you are better able to solve problems that involve rationalizing denominators of fractions under cube radical expressions, much like the problem of \(\sqrt[3]{\frac{2}{9}}\). This involves understanding how to manipulate and simplify the expression so that it becomes more straightforward and easier to work with.

Fractions

Fractions represent parts of a whole and consist of two numbers: the numerator, which is the top part, and the denominator, which is the bottom part. In the fraction \(\frac{2}{9}\), 2 is the numerator and 9 is the denominator. They express a ratio or a division of the numerator by the denominator, conveying how many parts we have of a given size.Working with fractions often involves operations including multiplication, division, addition, and subtraction, alongside techniques like finding common denominators.

• Fractions express parts of a whole, with the numerator on top and the denominator on the bottom.
• Operations with fractions are crucial to solving mathematical problems that involve ratios, proportions, and rational numbers.
• Understanding fractions also strengthens comprehension of more complex mathematical concepts, such as rationalizing expressions.

In the context of the given exercise, knowing how to multiply the numerator and denominator to rationalize \(\sqrt[3]{\frac{2}{9}}\) helps in simplifying the expression towards obtaining an integer under cube roots. It requires one to understand how fractions behave under multiplication and division.

Mathematical Operations

Mathematical operations include a variety of basic functions that we perform on numbers. These include addition, subtraction, multiplication, and division. They are the foundation of all calculations in mathematics. When rationalizing denominators, especially with fractions under cube roots, multiplication becomes a key operation. This involves multiplying the numerator and the denominator by the same number, a concept rooted in making the denominator a perfect power that removes the radical.

• Operations like multiplication help in adjusting numerators and denominators to rationalize.
• The same number is used to multiply both parts of a fraction to keep the expression equivalent.
• Simplifying results often involves mathematics operations to achieve a cleaner expression.

In the problem \(\sqrt[3]{\frac{2}{9}}\), multiplying both parts by 3 ensures that the denominator is itself a cube, making it easier to handle since \(9 \times 3 = 27\), which beautifully integrates into simplifying cube root expressions.