Problem 18 - Technical Mathematics with Calculus

Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Powers by Calculator. $$(1.35)^{5}$$

Step-by-Step Solution

Answer

The fifth power of 1.35, rounded to three significant digits, is approximately 4.23.

Step 1: Understanding the Problem

We are required to calculate the fifth power of 1.35 while maintaining the proper number of significant digits. Since 1.35 has three significant digits, our answer should also reflect that level of precision.

Step 2: Using a Calculator

Enter 1.35 into the calculator and then raise it to the power of 5. This can usually be done by pressing the '^' button or using the 'power' function, depending on your calculator's model.

Step 3: Rounding the Answer

After obtaining the calculated value, round it to three significant digits. This is because the original number, 1.35, has three significant digits.

Step 4: Presenting the Final Answer

Once you have the rounded value, this is the final answer that retains the proper number of significant digits.

Key Concepts

Powers by Calculator

Calculating powers might seem daunting at first, but modern calculators make this task much simpler. When you’re asked to compute something like \(1.35^{5}\) using your calculator, you're going through the process of raising a number to a particular power - in this case, 5.

To do this, you'll typically use either the '^' button or, if available, a dedicated 'power' function. Different calculators have slightly different methods to enter this computation. For standard scientific calculators, this process generally involves typing '1.35' followed by the power function and then '5'.

Some advanced calculators may also allow you to visualize the entire expression before computing, which can be very helpful in minimizing input errors. Always make sure to double-check your entry before pressing the '=' button.

Rounding Significant Digits

In mathematics and science, it's essential to present your answer with the correct number of significant digits to exhibit the right level of precision. After calculating \(1.35^{5}\) on a calculator, you will likely encounter a long string of digits. However, you must round this to the number of significant digits reflective of the initial value.

In our example, \(1.35\) has three significant digits, which means the final answer should too. To round the calculated result, look at the fourth digit after the decimal point: if it's 5 or higher, you would increase the third digit by one. If it's lower than 5, leave the third digit as is and discard the rest. Remember, zeros that merely serve as place holders are not considered significant.

Calculator Usage for Powers

Correct calculator usage is critical for arriving at the accurate result, especially when dealing with powers. After inputting the base number into your calculator, you must confirm that no additional operations unintentionally alter the value before raising it to the power. Ensuring that parentheses are correctly used is one way to prevent such errors.

For example, when entering \(1.35^{5}\) without parentheses, your calculator might interpret this as 1.35 times 5, not 1.35 raised to the power of 5. This illustrates why understanding how your specific calculator interprets inputs is so crucial. Always refer to the user manual for clarification if needed. After the power calculation, the result should appear on your screen, ready to be rounded accordingly.

Mathematical Precision

Mathematical precision is not just about getting the right answer; it's about representing that answer with the appropriate degree of exactness. This applies to every step of the calculation, from how you enter numbers into a calculator to how you round off the final answer.

When raising powers in calculations like \(1.35^{5}\) and rounding to the correct number of significant digits, you're practicing precision. Precision ensures that your result most closely corresponds to the actual value under the given constraints. This kind of rigor is crucial, especially in scientific fields where even the smallest deviation can have impactful consequences.

As a student, consistently applying the rules of significant digits helps to cultivate a habit of precision that is invaluable in professional scientific and mathematical work.