Problem 32 - A Graphical Approach to Precalculus with Limits

Question

Use a calculator to find each root or power. Give as many digits as your display shows. $$\sqrt[5]{-3}$$

Step-by-Step Solution

Answer

The fifth root of \(-3\) is approximately -1.24573.

Step 1: Understand the Expression

The expression given is \( \sqrt[5]{-3} \). This means we need to find the number that, when multiplied by itself five times, equals \(-3\).

Step 2: Calculate Using a Calculator

Use a scientific calculator and input \(-3^{1/5}\) to find the fifth root of \(-3\). Make sure the calculator is set to the correct mode to handle negative bases for non-integer exponents. The calculator will provide the result of \(\sqrt[5]{-3}\).

Step 3: Record the Result

Once the computation is complete, note down the number shown on the calculator's display. This number is the fifth root of \(-3\) to as many decimal places as the calculator provides.

Key Concepts

Scientific Calculator

A scientific calculator is a type of electronic calculator designed to calculate problems in science, engineering, and mathematics. Unlike a basic calculator, a scientific calculator can handle complex functions, such as exponentiation, roots, logarithms, and trigonometric functions. These are crucial tools for students and professionals who deal with roots and powers frequently.

• To operate a scientific calculator, first ensure it's in the correct mode for your calculation. For roots and powers, especially involving negative numbers, you may need specialized modes that allow for precise calculations.
• When dealing with fractional exponents, remember that scientific calculators can process these directly, such as using \(x^{y}\) or \(\sqrt[y]{x}\).
• Modern calculators may require you to input the fractional exponent as a decimal, e.g., \(1/5\) for a fifth root, ensuring it can manage the operation even with negative inputs.

Always remember to double-check your mode and settings to ensure accurate results, particularly with negative bases.

Fifth Root

The concept of the fifth root involves identifying a number which, when raised to the power of five, returns the original number. In mathematical terms, the fifth root of a number \(-3\) is written as \(\sqrt[5]{-3}\).

• To calculate the fifth root, consider the number you need to multiply five times to return to your initial value. In this case, the challenge arises due to the negative sign, which affects the calculation method.
• The fifth root can also be expressed as using powers, \((-3)^{1/5} \), which suggests taking the number to the \(1/5\) power.

When handling negative numbers, you must be mindful that roots of odd degree, like the fifth root, are defined for negative values. This differs from even degree roots, like square or fourth roots, which are not defined for negative numbers.

Negative Numbers

Negative numbers are real numbers that represent a value less than zero. In mathematics, they present unique characteristics, especially when it comes to operations involving powers and roots.

• With negative numbers, odd roots like the fifth root can yield real numbers as results. This is because multiplying a negative number an odd number of times results in a negative product.
• For example, to find the fifth root, solving \(x^{5} = -3\) is feasible as odd roots of negative numbers are negative since \((-x)^{5} = -(x^5)\).
• This is unlike even roots which do not allow negative results due to the squared terms yielding positive outcomes.

Furthermore, scientific calculators must be capable of interpreting such inputs correctly and providing the real results needed for analysis, ensuring they manage signs throughout the calculations.