Question

find the best decimal approximation that your calculator allows. Begin by making a mental estimate $$ \sqrt[4]{\left(6 \pi^{2}-2\right) \pi} $$

Short answer

The best calculator approximation for the expression is approximately 3.625.

Step-by-step solution

Understand the Expression

The expression given is the fourth root of \((6\pi^{2} - 2)\pi\). This means we need to compute \((6\pi^{2} - 2)\pi \) first and then find its fourth root.

Estimate \(\pi\)

We know \(\pi\) is approximately 3.14. So, \(\pi^{2} \approx (3.14)^2 \approx 9.86\).

Compute \(6\pi^{2} - 2\)

Using the estimated \(\pi^{2}\), we calculate: \(6 \times 9.86 - 2 \approx 59.16 - 2 = 57.16\).

Compute \((6\pi^{2} - 2)\pi\)

Multiply the result from step 3 by \(\pi\): \(57.16 \times 3.14 \approx 179.4624\).

Find the Fourth Root

Now take the fourth root of approximately 179.4624. Using a calculator, you find \(\sqrt[4]{179.4624} \approx 3.55\).

Best Approximation from Calculator

Input the expression directly into a scientific calculator to check: Find the fourth root of the expression directly, ensuring all calculations are to the maximum precision your calculator allows. The best approximation is \(3.624923\).

Key concepts

Mathematical Expression

Understanding a mathematical expression is like breaking down a sentence into understandable parts. In this exercise, the expression \( \sqrt[4]{(6 \pi^{2} - 2) \pi} \) involves a sequence of mathematical operations that need to be followed in a precise order. It begins with an inside calculation, such as dealing with powers of \( \pi \), then handling multiplications or subtractions, and finally applying the fourth root operation. To start, look at the core operation within the parentheses—calculate \( 6 \pi^{2} - 2 \). This means you need to increase \( \pi \) to the power of 2 first, multiply by 6, and then subtract 2 from the result. Recognizing and following these steps accurately ensures a correct result. Grasping this mathematical flow from inside to out is crucial and typical of handling such expressions.

Fourth Root Calculation

The concept of taking a fourth root might seem tricky at first glance. It is about finding a number which, when multiplied by itself four times, gives the original number. In mathematical terms, for a number \( x \), if \( x^4 = a \), then \( x \) is the fourth root. In this exercise, once the expression \((6 \pi^{2} - 2) \pi\) is simplified to a value such as approximately 179.4624, taking the fourth root is the next step. Using the formula \( \sqrt[4]{x} \), you need a calculator due to the complex nature of the number. The calculator will perform the operation swiftly to find that the fourth root results in about 3.55. Understanding the mathematical definition and calculator operation will make the process smoother.

Pi Approximation

Pi (\( \pi \)) is an irrational number approximately equal to 3.14159, often simplified to 3.14 in basic calculations for easier handling. This is a critical part of many mathematical expressions, especially those dealing with circles, spheres, and periodic functions.The approximation of \( \pi \) as 3.14 allows us to quickly estimate values in expressions where exact precision isn't mandatory. In this exercise, with \( \pi^{2} \approx 9.86 \), we see how approximations can help navigate through calculations efficiently. Still, for higher precision, especially in advanced mathematics and physics problems, using a more precise value or a calculator that includes \( \pi \) will give far better results. Recognizing when to use a close approximation versus an exact figure is a valuable skill.

Scientific Calculator Usage

Scientific calculators are powerful tools that assist with performing complex operations automatically. In cases where expressions involve intricate calculations, like taking a fourth root, these calculators can ensure precision and efficiency.To tackle this problem completely, you can directly input the expression \( \sqrt[4]{(6 \pi^{2} - 2) \pi} \) into a scientific calculator. Modern calculators allow for direct entry of the expression as seen or decomposing segment by segment to manage it manually. By using parentheses effectively, ensuring correct order of operations, and utilizing functions such as power and root calculations, a scientific calculator gives a result which, in this case, is precisely 3.624923. Becoming proficient with these devices is vital for simplifying tasks and achieving accuracy in mathematical problem-solving.