Problem 11 - Elementary and Intermediate Algebra

Question

Simplify by taking the roots of the numerator and the denominator. Assume that all variables represent positive numbers. $$ \sqrt[3]{\frac{64}{27}} $$

Step-by-Step Solution

Answer

\ \[ \frac{4}{3} \]

Step 1 - Identify the numerator and denominator

The expression given is \ \ \[ \sqrt[3]{\frac{64}{27}} \] Here, the numerator is 64 and the denominator is 27.

Step 2 - Take the cube root of the numerator

Find the cube root of 64. The cube root of 64 is \ \[ \sqrt[3]{64} = 4 \] because \ \[ 4^3 = 64. \]

Step 3 - Take the cube root of the denominator

Find the cube root of 27. The cube root of 27 is \ \[ \sqrt[3]{27} = 3 \] because \ \[ 3^3 = 27. \]

Step 4 - Form the simplified fraction

After calculating the cube roots, the fraction simplifies to \ \[ \frac{4}{3} \]

Key Concepts

Cube Roots

When simplifying radicals, especially cube roots, it's important to understand what a cube root is. A cube root is a number that, when multiplied by itself three times, gives the original number. For example, the cube root of 64 is 4 because when you multiply 4 by itself three times (4 × 4 × 4), you get 64. Similarly, the cube root of 27 is 3 because 3 × 3 × 3 equals 27. Finding the cube root simplifies expressions that contain roots raised to the power of three, making it easier to handle in fraction form or other mathematical operations.

Numerator and Denominator

In any fraction, the top number is called the numerator, and the bottom number is the denominator. In the given expression \(\frac{64}{27}\), 64 is the numerator and 27 is the denominator. The numerator indicates how many parts of a whole you have, and the denominator shows how many equal parts the whole is divided into. When simplifying a fraction that involves cube roots, you take the cube root of both the numerator and the denominator separately. This process helps in breaking down the components into simpler form which can then be easily worked with.

Fraction Simplification

Fraction simplification is a key mathematical skill. It involves reducing a fraction to its simplest form so that it is easier to work with or interpret. In our case, we simplified \(\frac{64}{27}\) by taking the cube root of both the numerator and the denominator. First, we found that \(\frac{64}{27}\) could be rewritten as \[ \frac{\root{3}{64}}{\root{3}{27}} \], which further simplifies to \[ \frac{4}{3} \]. This means that the fractional value of the original expression can be more easily understood as four-thirds, which is 1 and 1/3 in mixed number form. Simplifying fractions aids in clear communication of mathematical ideas and helps in further calculations or applications.