Problem 36 - A Graphical Approach to College Algebra

Question

Use a calculator to find each root or power. Give as many digits as your display shows. $$\sqrt[5]{-8}$$

Step-by-Step Solution

Answer

The fifth root of \\(-8\\) is approximately \\(-1.5157165665\\).

Step 1: Understand the Expression

We are asked to find the fifth root of \(-8\). The expression \(\sqrt[5]{-8}\) means finding a number, which when multiplied by itself four more times (for a total of five times), results in \(-8\).

Step 2: Set Up the Calculation

On the calculator enter \(-8\) and use the root function or power key, often denoted as \(x^{\frac{1}{5}}\) or a similar function on scientific calculators. This will calculate the fifth root of the number.

Step 3: Perform the Calculation

Enter the value \(-8\), access the function for the fifth root, and complete the calculation. Your calculator should display the result as \(-1.5157165665\) based on the precision available.

Step 4: Verify the Result

To confirm the fifth root, verify by raising \(-1.5157165665\) to the power of 5. The result should be approximately \(-8\), within any slight variations due to calculator precision.

Key Concepts

Understanding Scientific Calculators

Scientific calculators are essential tools for solving complex mathematical operations. They are more advanced than basic calculators and include functions necessary for calculus, statistics, and more. For students learning about roots and exponents, understanding how to use a scientific calculator is crucial.

• These calculators come equipped with a variety of functions like powers, roots, logarithms, and trigonometric calculations.
• Keys are often labeled with symbols such as \(x^{\frac{1}{n}}\) to denote roots or exponents.
• They can handle both positive and negative numbers, making them versatile for different math problems.

To find the fifth root of \(-8\) using a scientific calculator, input the number and then use the correct function to perform the operation. Ensure you know how to access these specialized functions, as it's slightly different depending on the calculator model.

Exploring Rational Exponents

Rational exponents are a way of expressing roots as exponents. This notation can simplify calculations and makes the relationship between powers and roots clearer.

• A rational exponent like \(x^{\frac{1}{5}}\) represents the fifth root of \(x\).
• It combines the ideas of powers and roots into one expression.
• The rational exponent \(\frac{1}{5}\) suggests that the number needs to be multiplied by itself four more times to reach the original number.

By using rational exponents, calculations become easier, especially when using calculators. For instance, entering \(-8^{\frac{1}{5}}\) can directly show the fifth root of the number.

Achieving Calculator Precision

Calculator precision is the degree of exactness with which a calculator's display shows a number. With scientific calculators, precision varies depending on the model and settings.

• Most calculators offer anywhere from 8 to 12 decimal places for results.
• This precision is crucial for math problems requiring detailed solutions.
• It’s essential to understand that small discrepancies might occur due to rounding or approximation.

The calculated fifth root, \(-1.5157165665\), reflects the maximum precision the calculator can provide. While this is sufficient for most practical purposes, it's good to be aware that more precise tools might be needed for complex scientific calculations.