Question

Use a calculator to evaluate the expression. Round your result to three decimal places.\(6^{-\sqrt{2}}\)

Short answer

The result of the expression, rounded to three decimal places, will depend on the calculator used. However, it should approximately equal 0.252.

Step-by-step solution

Calculate the Square Root

The first operation to perform in the given expression is to calculate the square root of 2. Scientific calculators typically have a square root function denoted by '\(\sqrt{}\)'. Use this function to find the square root of 2.

Apply the Exponent

Next, use the result of Step 1 as the exponent of 6. On your calculator, you can usually do this by entering '6', then hitting the '^' button, then entering your result from Step 1. This will give you \(6^{-\sqrt{2}}\).

Round to Three Decimal Places

The final step is to round the result from Step 2 to three decimal places. Make sure to only round your final result instead of rounding intermediate steps, which could introduce rounding errors.

Key concepts

Square Root

The square root is a fundamental mathematical operation that determines which number, when multiplied by itself, equals the original number. For example, the square root of 4 is 2, because 2 times 2 equals 4. Mathematically, this is represented as \(\sqrt{4} = 2\).

In practice, calculating a square root can often be done with a scientific calculator. Most calculators feature a square root button, usually represented by the symbol \(\sqrt{}\). By entering the number you wish to find the square root of and pressing this button, you can quickly determine the square root value.

For this exercise, you need to calculate the square root of 2. When you press the square root button after entering the number 2 on your calculator, you will get approximately 1.414. This value is critical as it is used in the next step of applying an exponent.

Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in a compact form. It is particularly useful in science and engineering. Numbers are written in the form of \(a \times 10^b\), where \(a\) is a number greater than or equal to 1 and less than 10, and \(b\) is an integer.

Understanding how to use scientific notation on a calculator can come in handy when dealing with large exponents, such as in the expression \(6^{-\sqrt{2}}\). After computing operations like powers, results can be very large or very small. Rather than writing out numerous zeroes, scientific notation offers a simpler representation.

For instance, the number 0.000123 can be written as \(1.23 \times 10^{-4}\). This expression is easier to read and work with, especially in calculations.

Rounding Decimals

Rounding decimals is the process of reducing the number of significant digits in a number while keeping the representation as accurate as possible. This is crucial in many mathematical and scientific calculations where complete precision is unnecessary or difficult to maintain.

To round a number to a specific decimal place, you look at the digit following the place you are rounding to:

• If the next digit is 5 or greater, you round up.
• If it is less than 5, you round down.

For example, if you want to round 3.146 to two decimal places, you look at the third decimal, which is 6, and round up the second decimal to get 3.15.

In the context of the given expression, after computing \(6^{-\sqrt{2}}\), you will obtain a numerical result which might have many digits. You need to round this result to three decimal places as per the requirement of the problem, ensuring clarity and precision in your answer.