Problem 17 - Intermediate Algebra

Question

Evaluate each of the following. For example, \(\sqrt{25}=5\). \(\sqrt[3]{\frac{27}{64}}\)

Step-by-Step Solution

Answer

The value of \(\sqrt[3]{\frac{27}{64}}\) is \(\frac{3}{4}\).

Step 1: Identify the Expression

We need to evaluate the expression \(\sqrt[3]{\frac{27}{64}}\). This is a cube root expression, which means we're looking for a number that, when multiplied by itself three times, equals \(\frac{27}{64}\).

Step 2: Separate the Numerator and Denominator

We separate the cube root into two parts: the cube root of the numerator and the cube root of the denominator. This gives us: \(\sqrt[3]{27}\) and \(\sqrt[3]{64}\).

Step 3: Evaluate the Cube Root of the Numerator

Evaluate \(\sqrt[3]{27}\). Since \(3 \times 3 \times 3 = 27\), the cube root of 27 is 3.

Step 4: Evaluate the Cube Root of the Denominator

Evaluate \(\sqrt[3]{64}\). Since \(4 \times 4 \times 4 = 64\), the cube root of 64 is 4.

Step 5: Combine the Results

Combine the results from both the numerator and the denominator: \(\frac{\sqrt[3]{27}}{\sqrt[3]{64}} = \frac{3}{4}\).

Key Concepts

Numerator and Denominator Operations

When dealing with fractions in expressions, it's crucial to handle the numerator and the denominator separately. In the case of cube roots, like in the expression \(\sqrt[3]{\frac{27}{64}}\), we treat the numerator (27) and the denominator (64) individually. This separation simplifies our calculations.By evaluating \(\sqrt[3]{27}\) and \(\sqrt[3]{64}\) independently, we break down the problem into manageable parts:

• For the numerator \(27\), we find that \(3 \times 3 \times 3 = 27\), giving the cube root as 3.
• For the denominator \(64\), because \(4 \times 4 \times 4 = 64\), the cube root is 4.

These operations allow us to rewrite the expression as \(\frac{3}{4}\). By handling each component separately, complex fractional cube roots become simpler to evaluate.

Fraction Simplification

Once you have evaluated the cube roots of both the numerator and the denominator separately, the next step is fraction simplification. Fraction simplification involves reducing a fraction to its simplest form by dividing both the numerator and the denominator by their greatest common divisor (GCD). In our example, after calculating \(\sqrt[3]{27} = 3\) and \(\sqrt[3]{64} = 4\), we have the fraction \(\frac{3}{4}\). The numbers 3 and 4 have no common divisors other than 1, so the fraction is already in its simplest form.Simplifying fractions is important as it makes the numbers easier to understand and compare. It is always a good habit to check if your fraction can be simplified further by looking for common divisors.

Algebraic Expressions

Algebraic expressions encompass operations such as addition, subtraction, multiplication, division, and also include roots. Solving expressions with cube roots is a fundamental skill in algebra. The expression \(\sqrt[3]{\frac{27}{64}}\) is an example of how understanding operations in algebra can simplify solutions of seemingly complex equations.In algebraic terms:

• Expressions represent numbers or quantities.
• Operations like cube roots are used to find numbers that, when raised to a power, return another specified number (like finding a number that cubed returns 27 or 64).

By breaking down an expression into its algebraic components, you can apply operations systematically. Learning to work with these components means even intricate algebraic problems become easier to tackle. Mastery of these expressions is important as they appear frequently in advanced mathematics and various real-world applications.