Problem 35 - 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. $$\sqrt[3]{-17}$$

Step-by-Step Solution

Answer

The cube root of -17 is approximately -2.57128, based on typical calculator precision.

Step 1: Understand the Expression

The expression \( \sqrt[3]{-17} \) asks for the cube root of -17. This means we are looking for a number which, when multiplied by itself twice, gives -17.

Step 2: Use Calculator Functions

Enter -17 into the calculator. Use the cube root function, often represented as \( x^{1/3} \) or a specific button depending on the calculator model, to compute the cube root.

Step 3: Interpret the Display

Once the cube root function has been applied, observe the number shown on the calculator display. Write down all the digits as displayed, as these represent the calculation's precision.

Key Concepts

Calculator Functions

When tasked to find the cube root of a number, especially a negative one like -17, your calculator becomes an essential tool. Let's explore how to properly use calculator functions for such calculations.

• First, ensure your calculator is scientific as not all basic calculators can compute cube roots.
• Input the target number, in this case, -17, and look for a button labeled either as the cube root symbol \( \sqrt[3]{} \) or an exponent button \( x^{y} \).
• In some calculators, to input a cube root, you might need to raise the number to the power of one-third \( x^{(1/3)} \).

After entering the number -17, press the corresponding function or sequence to get the cube root. Calculators may differ, so refer to your calculator's manual if unsure. Remember, using these functions efficiently can simplify complex calculations and improve accuracy.

Interpreting Calculator Display

Once you've entered the command to find the cube root of -17, your calculator will display the result. Interpreting this display correctly is crucial for understanding and using the answer.

• Review the digits shown on the display; they represent the precise value of the cube root to the extent of your calculator's capability.
• Realize that a long sequence of digits indicates a non-terminating decimal, a common case when dealing with irrationals.
• Write down the entire sequence of numbers as it appears, because rounding prematurely can lead to inaccuracies in further calculations.

Pay attention to whether the calculator uses parentheses, commas, or a specific notation to represent very large or very small numbers. This can help in interpreting results precisely, especially with scientific calculators.

Exponents

Understanding exponents is central to grasping cube roots and other root functions. Let's look deeper into the role exponents play:

• An exponent represents how many times you multiply a number by itself. For example, in \( x^3 \), the number \( x \) is multiplied by itself twice (\( x \times x \times x \)).
• When calculating cube roots, you're essentially looking for a number that produces the original value when raised to the third power.
• The exponent \( \frac{1}{3} \) is the mathematical notation used to represent the cube root.

Grasping the relationship between exponents and roots can simplify your understanding of not just cube roots, but also higher-order roots and their calculations.