Problem 37 - 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. $$\sqrt[6]{\pi^{2}}$$

Step-by-Step Solution

Answer

1.27457110896

Step 1: Calculate the Power

First, calculate \(\pi^{2}\).Using a calculator, exponentiate \pi (approximately 3.14159265359) to the power of 2.This results in \(\pi^{2} \approx 9.86960440109\).

Step 2: Take the Sixth Root

Next, take the sixth root of the value obtained in Step 1.Using a calculator, input \(9.86960440109\) and raise it to the power of \(\frac{1}{6}\).This calculates the sixth root, which results in approximately \(1.27457110896\).

Key Concepts

Exponents

Understanding exponents is key in solving problems involving roots and powers, as they tell us how many times to multiply a number by itself. When working with exponents, you may see expressions such as \( x^n \), where \( x \) is called the base and \( n \) is the exponent. This means you multiply \( x \) by itself \( n \) times. Let's consider the number \( \pi \) in this specific context. Here, \( \pi \approx 3.14159265359 \), and we need to find \( \pi^2 \), which involves multiplying \( \pi \) by itself. The calculation is:

• \( \pi^2 = \pi \times \pi = 3.14159265359 \times 3.14159265359 \)
• This gives \( \pi^2 \approx 9.86960440109 \)

Exponents are not restricted to whole numbers; they can also be negative or fractions, indicating division or roots, respectively. This flexibility is vital in more advanced mathematical operations, as we will see with the sixth root.

Sixth Root

Taking the sixth root of a number is the reverse operation of raising a number to the sixth power. When you calculate the sixth root of a number such as \( x \), you are essentially looking for a number \( y \) such that \( y^6 = x \). In our exercise, after determining \( \pi^2 \approx 9.86960440109 \), the task is to find the sixth root of this result. To do this, you can use the property that taking the \( n \)th root of a number is equivalent to raising that number to the power of \( 1/n \):

• Expressed as \( x^{1/6} \)
• For our calculation: \( 9.86960440109^{1/6} \)
• Which is approximately \( 1.27457110896 \)

This highlights the interconnected nature of exponents and roots, emphasizing that understanding one aids in comprehending the other.

Calculator Use

Using a calculator effectively is crucial for solving complex math problems quickly and accurately. Calculators can easily perform operations such as exponentiation and root extraction. When handling tasks involving exponents and roots, ensure that your calculator is in the correct mode (degree, radian, or as required) and that you are familiar with using its functions. Here are some tips:

• **Finding powers**: Input the base, press the exponent function (often labeled as \( x^y \)), and enter the exponent value.
• **Extracting roots**: Most calculators allow roots to be calculated by using the exponent function with a fractional power (e.g., for sixth root, use \( y^{1/6} \)).
• Always double-check your calculations, especially when entering several digits, like with \( \pi \).

These skills will help you use your calculator more proficiently, making solving equations involving roots and powers much more manageable.