Problem 61 - A Graphical Approach to College Algebra

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: Understand the Expression

The expression given is \( 19^{1/2} \), which means we need to find the square root of 19. This is the same as finding \( \sqrt{19} \).

Step 2: Use a Calculator

To find the square root of 19, we can use a calculator because it is not a perfect square and does not result in an exact integer. Input \( 19 \) into the calculator and find \( \sqrt{19} \).

Step 3: Analyze the Result

After calculating \( \sqrt{19} \), the result is approximately 4.358898944. When rounding to the nearest thousandth, consider the fourth decimal place.

Step 4: Round the Result

The number obtained is 4.358898944. To round this to the nearest thousandth, look at the fourth decimal place, which is 8. Since it is greater than 5, we round up the third decimal place, resulting in 4.359.

Key Concepts

Decimal Approximation

When dealing with mathematical calculations, especially those involving roots and powers, decimal approximation becomes an invaluable tool. An approximation is a number close to the actual value. For square roots that don't have a neat answer, like in the case of \( \sqrt{19} \), we use decimal approximations to express the result.

• In our example, the exact value of \( \sqrt{19} \) isn't a simple integer, so we need a decimal format.
• Using a calculator, we find \( \sqrt{19} \approx 4.358898944 \).
• This is the decimal approximation of \( \sqrt{19} \), helpful when precision is needed.

A key takeaway here is that although approximations aren't always perfect, they offer significant precision that's often more useful in practical settings than a purely symbolic representation.

Exponents

In mathematics, exponents play a crucial role in expressing repeated multiplication. An exponent indicates how many times a number, known as the base, is multiplied by itself. For instance, \( 3^2 \) means \( 3 \times 3 \).

• In the given problem, \( 19^{1/2} \), the exponent is \( \frac{1}{2} \).
• This particular exponent signifies a square root operation, as taking a number to the power of \( \frac{1}{2} \) is the same as finding its square root.
• Thus, \( 19^{1/2} \) is simply another way of expressing \( \sqrt{19} \).

Understanding exponents as roots can be empowering, as it reveals the dual nature of these operations, paving the way for solving more complex equations seamlessly.

Rounding Numbers

Rounding is a mathematical technique used to reduce a number to a less precise number of digits. It simplifies calculations and makes results more digestible when exact precision isn't necessary.

• In rounding, we focus on a specific place value - in this case, the nearest thousandth or three decimal places.
• To determine if we should round up or down, we look at the digit immediately following our target decimal place in the original number.
• As seen in the example, \( 4.358898944 \), we focus on the fourth decimal digit: 8.

Since 8 is greater than 5, we round the third decimal place from 8 to 9, resulting in \( 4.359 \). Rounding in this way helps keep our answers manageable and easier to communicate.