Problem 75 - A Graphical Approach to College Algebra

Question

Approximate each expression to the nearest hundredth. $$\frac{2}{1-\sqrt[3]{5}}$$

Step-by-Step Solution

Answer

The approximate value is -2.82.

Step 1: Identify the Expression

We need to approximate the expression \( \frac{2}{1-\sqrt[3]{5}} \) to the nearest hundredth. The expression involves a cube root in the denominator.

Step 2: Approximate the Cube Root

Calculate \( \sqrt[3]{5} \) using a calculator. This value is approximately \( 1.71 \).

Step 3: Substitute the Approximate Value

Replace the cube root in the expression with its approximate value: \( \frac{2}{1-1.71} \).

Step 4: Simplify the Expression

Calculate the denominator: \( 1 - 1.71 = -0.71 \). Thus, the expression becomes \( \frac{2}{-0.71} \).

Step 5: Perform Division

Calculate \( \frac{2}{-0.71} \) using a calculator, which gives approximately \( -2.82 \).

Step 6: Round to the Nearest Hundredth

The result from the previous step is already at the nearest hundredth, \( -2.82 \).

Key Concepts

Approximation

When we talk about approximation in mathematics, we are referring to finding a value that is close to the exact number, but easier to use or understand. Approximating is handy when exact values are difficult to work with. For instance, when calculating cube roots or dealing with irrational numbers.

In our exercise, the goal is to approximate the expression \( \frac{2}{1-\sqrt[3]{5}} \) to the nearest hundredth. This means rounding our final result to two decimal places, making the number easier to interpret and utilize in further calculations.

While performing approximations, calculators are often used to provide a result that is close to the true value, enhancing our ability to quickly solve mathematical problems.

Cube root

A cube root is a number that, when multiplied by itself twice, gives the original number. For example, the cube root of 8 is 2, since \(2 \times 2 \times 2 = 8\). Cube roots can be more complex when dealing with numbers that are not perfect cubes, which is the case with 5.

Finding the cube root of numbers like 5 often requires a calculator, as they do not result in neat whole numbers. For this exercise, we approximated \( \sqrt[3]{5} \) to be about \(1.71\). This approximation helps simplify our calculations since working with such a decimal is easier than handling its exact cube root form.

Understanding cube roots and how to calculate them is crucial in solving advanced algebraic expressions, especially when simplifying denominators or other mathematical parts involving roots.

Denominator simplification

Denominator simplification comes into play when we need to make a fraction easier to calculate or understand. In our expression \( \frac{2}{1-\sqrt[3]{5}} \), the denominator \(1-\sqrt[3]{5}\) is not simple to work with directly, due to the cube root.

After approximating the cube root of 5 to \(1.71\), we substitute this value back into the expression, changing the denominator to \(1 - 1.71\). This subtraction yields \(-0.71\) as the new denominator. Simplifying this denominator allows us to proceed with the division easier, producing a sensible approximation of the entire expression.

The final step is to complete the division, \(\frac{2}{-0.71}\), achieving a simplified fractional result that can then be rounded to the nearest hundredth for clarity and ease of use. Mastering the skill of denominator simplification aids in effectively solving a multitude of mathematical problems.