Problem 75 - College Algebra and Calculus: An Applied Approach

Question

Use a calculator to approximate the number. (Round to three decimal places.)\(5.7^{2 / 5}\)

Step-by-Step Solution

Answer

The approximate number is 2.315

Step 1: Identify the base and exponent

In \(5.7^{2 / 5}\), 5.7 is the base and \(2 / 5\) or 0.4 is the exponent.

Step 2: Input in the calculator

Input the number with the fractional exponent into the calculator. This usually means typing '5.7', then the exponent button (often marked '^'), and finally '2 / 5' or '0.4'.

Step 3: Calculate and round

Calculate the value and round the answer to three decimal places. You will find the 'rounding' function on most scientific calculators.

Key Concepts

Fractional exponents

Fractional exponents can seem intimidating at first, but once you understand them, they're quite logical. An exponent tells you how many times to multiply a number by itself. When dealing with fractional exponents, you're essentially doing two things:

• The numerator of the fraction indicates the power to which the base is raised.
• The denominator determines the root of the base that you are taking.

So for the expression like \(5.7^{2/5}\), you first raise 5.7 to the power of 2 (which is squaring it), and then take the fifth root of the result. Another way to think about this operation is that you are looking for a number that when raised to the 5th power, will equal \(5.7^2\). Understanding that a fractional exponent is a combination of power and root can make complex expressions more manageable.

Calculator usage

Using a calculator for expressions involving exponents can save time and increase accuracy, especially as calculations become more complex. Here’s a quick guide on how to use a calculator for fractional exponents:

• First, make sure you are familiar with your calculator's functionalities. Look for the exponent button, which is often represented by '^'.
• To calculate \(5.7^{2/5}\), first input 5.7.
• Next, press the exponent button.
• Finally, enter the fraction. Many calculators require you to enter the fraction as a decimal, 0.4 in this case. Alternatively, some advanced calculators allow you to input the fraction directly.
• After inputting, hit the calculate/equals button to find the result.

Ensuring that you input numbers and operations correctly is crucial for achieving the right answer.

Rounding decimals

Rounding is a fundamental skill in math that can help you simplify numbers, making them easier to work with. When dealing with decimals, rounding to a specific place can help standardize your results:

• Identify the decimal place you need to round. For three decimal places, it's the number in the thousandths place.
• Look at the number immediately to the right. If this number is 5 or greater, increase the target decimal place by one. If it's less than 5, keep the target decimal place the same.
• Round off any numbers to the right of the target place.

So, if your calculator gives you a result of 1.4428 for \(5.7^{2/5}\), you would look at the fourth decimal place (8), increase the third decimal place (2) by one, resulting in 1.443. With practice, rounding becomes second nature, helping keep your solutions neat and precise.