Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Powers by Calculator. $$(1.94)^{7}$$

Short answer

233. (rounded to three significant digits)

Step-by-step solution

Understanding significant digits

Before calculating the power, understand that the number 1.94 has three significant digits. This is because all non-zero numbers are considered significant, and zeroes between non-zero numbers are also significant.

Calculate the power

Using a calculator, compute the seventh power of 1.94. You can usually do this by typing '1.94' followed by the power button (often marked as '^' or 'y^x' or similar), and then '7', and lastly press the equals button or 'enter'.

Rounding to significant digits

Your calculator will likely give you a long number. Round this answer to the proper number of significant digits, which is three. You should use the standard rounding rules: if the first non-significant digit is 5 or greater, round up the last significant digit; otherwise, leave it as is.

Key concepts

Powers by Calculator

Calculating powers using a calculator is a straightforward process but it is important to be familiar with the specific functions of your calculator. The process typically involves inputting the base number, in this case, 1.94, and then using the power button which could be labelled in various ways, such as '^', 'y^x', 'EXP', or a button with a superscripted x. After pressing the power button, you enter the exponent, which is 7 for this example, and finally, you would press the 'equals' or 'enter' button to get the result.

It's crucial to remember that the precision of your calculator's result might exceed the required significant digits. Understanding how to interpret and modify the result as per the significant digits is an essential skill in mathematics, and this is where the concept of significant digits plays a vital role.

Rounding Rules

Proper rounding is essential for maintaining the accuracy of your calculations while adhering to the significant digits required. The basic rounding rules are relatively simple: look at the digit immediately to the right of the last significant digit you want to keep. If this digit is 5 or higher, you increase the last significant digit by one. If it's less than 5, you leave the last significant digit unchanged. There are additional rules when dealing with zeroes, which help to maintain the value's precision.

For instance, in our exercise, after calculating \(1.94^7\), the calculator might show a long number. However, since 1.94 has three significant digits, we round the result accordingly. These rules ensure that the final answer reflects the precision of the given data, and every digit in your answer has real statistical or practical significance.

Scientific Notation

Scientific notation is a method of writing numbers that are too large or too small to be conveniently written in decimal form. It is often used in mathematics, the sciences, and engineering. The format is \( M \times 10^n \), where \( M \) is a number greater than or equal to 1 but less than 10, and \( n \) is an integer. For example, the number 0.0000194 can be written as \(1.94 \times 10^{-5}\) in scientific notation.

In the context of significant digits, scientific notation is particularly useful. It clearly shows which digits are significant by presenting the numbers as a product of a specific number \( M \) and a power of ten. This makes the process of maintaining the correctness of significant digits through multiple-step calculations much more manageable. When you are calculating powers or applying any operation on numbers expressed in scientific notation, it helps ensure that the final result is presented with the correct number of significant figures.