Problem 39 - 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. $$13^{-1 / 3}$$

Step-by-Step Solution

Answer

Use a calculator to find that the value of \(13^{-1/3}\) is approximately 0.484.

Step 1: Understand the Expression

The expression given is \(13^{-1/3}\). This means we are looking for the reciprocal of the cube root of 13.

Step 2: Find the Cube Root

First, we need to calculate the cube root of 13, which is expressed as \(13^{1/3}\). Use your calculator to find this value.

Step 3: Find the Reciprocal

Once you have the cube root of 13, take the reciprocal by dividing 1 by the cube root value you found.

Step 4: Use the Calculator

Enter \(13^{-1/3}\) directly into your calculator or use the result from previous steps to find the reciprocal. Ensure that your calculator is set to display as many digits as possible.

Step 5: Record the Answer

Check the display of your calculator for the numerical result and record it as the solution, making sure to include all the digits shown.

Key Concepts

Cube Roots

A cube root of a number is a value that, when multiplied by itself three times, yields the original number. For instance, the cube root of 27 is 3, because \(3 \times 3 \times 3 = 27\). Calculating the cube root is essentially the inverse operation of raising a number to the power of three. It is denoted as \(x^{1/3}\) for a number \(x\). When you're dealing with numbers like 13, which are not perfect cubes, you will need a calculator to find the cube root. Simply enter \(13^{1/3}\) into the calculator, and it will provide a decimal approximation. This is essential in exercises where precision is fundamental. Calculators give you the advantage of accuracy and speed in finding cube roots of non-perfect cubes.

Reciprocal Calculation

The concept of a reciprocal is vital in mathematics, especially when dealing with fractions and powers. A reciprocal of a number is simply 1 divided by that number. In symbolic form, the reciprocal of \(x\) is \(\frac{1}{x}\). So, if we have calculated the cube root of 13 to be approximately 2.3513 (or whatever your calculator displays), its reciprocal would be \(\frac{1}{2.3513}\). Calculators make it easy to find reciprocals, typically by using a dedicated button or function. This concept is crucial when dealing with negative exponents, as they represent the reciprocal of that number raised to the corresponding positive exponent. Therefore, \(x^{-n}\) is equivalent to \(\frac{1}{x^n}\). Understanding this simplifies complex calculations like our provided exercise.

Exponents in Precalculus

Exponents are a fundamental part of precalculus, representing how many times a number, known as the base, is multiplied by itself. For instance, in \(13^{-1/3}\), 13 is the base and \(-1/3\) is the exponent. An exponent that is a fraction, such as \(1/3\), indicates a root; in this case, the cube root. Meanwhile, a negative exponent indicates a reciprocal. Hence, \(13^{-1/3}\) represents the reciprocal of the cube root of 13. These operations are common in equations and functions within precalculus, making a solid understanding essential. Calculators often handle these calculations effectively, allowing students to focus on the application rather than manual computation. Knowing these rules is critical for solving complex expressions.

Step-by-Step Problem Solving

Solving problems in steps can make seemingly complex calculations more manageable. It involves breaking down the problem into smaller, more digestible parts, dealing with one operation at a time. In the exercise \(13^{-1/3}\), the step-by-step approach involves first understanding what each part of the expression signifies, then methodically calculating the cube root, followed by the reciprocal. Each step builds on the last, ensuring accuracy and clarity. This method not only simplifies the calculation process but also helps in reinforcing the fundamental concepts underlying the problem. Emphasizing step-by-step solutions enhances learning, especially with technological tools like calculators that complete each step quickly and accurately, helping focus on understanding rather than getting overwhelmed by the operations.