Problem 61 - A Graphical Approach to Precalculus with Limits

Question

Find a decimal approximation of each root or power Round answers to the nearest thousandth. $$19^{1 / 2}$$

Step-by-Step Solution

Answer

The decimal approximation of \( 19^{1/2} \) is 4.359.

Step 1: Identify the Power

Determine what mathematical operation is required for the expression \( 19^{1/2} \). Here, \( '19^{1/2}' \) means we want to find the square root of 19.

Step 2: Use Square Root Formula

Recognize that the expression \( 19^{1/2} \) can be computed using a calculator for better precision, as it represents \( \sqrt{19} \). As calculators are often more accurate, use one to find \( \sqrt{19} \).

Step 3: Calculate Using a Calculator

Input the expression \( \sqrt{19} \) or \( 19^{1/2} \) into a calculator to compute its value. The calculator will provide a decimal approximation.

Step 4: Round to the Nearest Thousandth

Look at the decimal number obtained from the calculator. If the number is something like 4.35889894, then we round it to the nearest thousandth (three decimal places). Thus, the rounded result is 4.359.

Key Concepts

Square Root

The square root is a mathematical operation that finds a number which, when multiplied by itself, gives the original number. In simpler terms, if you are looking for the square root of 19, you are searching for a number that, multiplied by itself, equals 19. To symbolize this operation, we often use the radical sign, \( \sqrt{} \), before the number. For example, \( \sqrt{19} \) represents the square root of 19.
If finding the exact square root of a non-perfect square like 19 seems tricky, don't worry. We use decimal approximations for these square roots because they don't result in whole numbers. Using a calculator is invaluable here, as it provides a closer approximation of such numbers.

Exponents

Exponents are a shorthand technique to express repeated multiplication of the same number by itself. But sometimes, they can also represent different operations, such as roots. When dealing with \( 19^{1/2} \), you can see this is not your typical exponent case. Instead, it denotes a square root operation. This specific notation, 19 raised to the 1/2 exponent, is mathematically equivalent to taking the square root of 19.

• Regular exponents like \( x^3 \) means \( x \times x \times x \).
• Fractional exponents, such as \( x^{1/2} \), often represent root operations.

Recognizing these variations helps in simplifying and understanding different mathematical expressions.

Rounding numbers

Rounding numbers is an essential technique for simplifying numbers in practical use, especially when dealing with lengthy decimal approximations. When you round a number, you keep it close to its original value while reducing the number of decimal places. This makes it easier to read and use in calculations. To round to the nearest thousandth, you need to focus on the fourth decimal place.Here's how you do it:

• Identify the digit in the thousandth place (third place after the decimal, like 4.358 in \( \sqrt{19} \)).
• Look at the digit directly to the right (fourth place after the decimal).
• If this digit is 5 or greater, increase the thousandth place digit by one; if it is less than 5, keep it the same.

This way, the decimal approximation becomes easier to handle, yet accurate enough for most practical purposes.