Problem 35 - A Graphical Approach to College Algebra

Question

Use a calculator to find each root or power. Give as many digits as your display shows. $$\sqrt[3]{-17}$$

Step-by-Step Solution

Answer

The cube root of \(-17\) is approximately \(-2.5713\).

Step 1: Identify the Expression

The given expression is the cube root of negative seventeen, denoted as \(\sqrt[3]{-17}\). This expression asks us to find a number which, when raised to the power of three, equals \(-17\).

Step 2: Input Into Calculator

Use a calculator that can compute cube roots. On most scientific calculators, you can find a function or button for cube roots, often labeled as \(\sqrt[3]{x}\) or similar. Input \(-17\) into this function.

Step 3: Calculate Cube Root

After inputting \(-17\), the calculator will compute the cube root. The result should be displayed on the screen with as many decimal places as the calculator allows.

Step 4: Write Down Result

Record the number shown on your calculator display, which is the cube root of \(-17\). For example, the calculator might display \(-2.5713\), depending on its precision capabilities.

Key Concepts

Scientific Calculator

A scientific calculator is an essential tool for many mathematical operations. Unlike basic calculators, scientific calculators come equipped with advanced functions, including the ability to calculate roots, powers, logarithms, and trigonometry functions. To find the cube root of a number like \(-17\), you need to use this advanced functionality.

Here's a simple guide on using a scientific calculator for cube roots:

• Locate the cube root function. It is often labeled as \(\sqrt[3]{x}\) or just a special key with a cube root symbol.
• Input the number for which you want the cube root. In this case, it's \(-17\).
• Press enter or the equal symbol to compute the result. The calculator will display the cube root, which might appear with several decimal places depending on your device's display capacity.

Using a scientific calculator efficiently can save time and improve accuracy in solving mathematical problems, as it allows one to focus more on understanding the concepts than on computational details.

Negative Numbers

Understanding negative numbers is crucial when working with cube roots. While finding the square root of a negative number without using imaginary numbers isn't possible in the realm of real numbers, cube roots can indeed be taken for negative numbers.

Here's why:

• When a negative number is raised to an odd power, the result remains negative.
• Conversely, the cube root of a negative number yields a negative result.

For instance, when computing \(\sqrt[3]{-17}\), the output is a negative number, approximately \(-2.5713\) as mentioned in the solution. This allows us to conclude that the number, when cubed, will indeed revisit the original \(-17\), thanks to the characteristics of negative numbers with odd powers.

Decimal Places

When using a scientific calculator to find cube roots, the precision of your answer depends heavily on the number of decimal places shown on the device. Decimal places are crucial for accuracy in mathematics, especially when dealing with irrational numbers.

For example:

• An answer with more decimal places is closer to the true value, which is particularly important in fields requiring high precision, like engineering and physics.
• Most scientific calculators display results up to 8-10 decimal places, ensuring a high level of detail.

In the case of \(\sqrt[3]{-17}\), rounding the result to different decimal places could slightly alter the outcome, affecting further calculations. Always write down as many decimal places as your calculator allows to prevent rounding errors unless a problem specifically instructs otherwise.