Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Powers by Calculator. $$(2.26)^{6}$$

Short answer

The result of \(2.26)^{6}\ is approximately 113.2 when rounded to three significant digits.

Step-by-step solution

Understand Significant Digits

Significant digits are the digits in a number that carry meaning towards its precision. When using calculators to evaluate powers, the result should retain the number of significant digits that is in the base number. For the base number 2.26, there are three significant digits.

Calculate the Power

Use a calculator to raise the base number 2.26 to the power of 6. Enter 2.26, then press the power (often represented by '^' or a 'y^x' button) button, and then enter 6.

Round the Result

After calculating, the calculator will give an answer. Make sure that your final answer has the same number of significant digits as the base number. As 2.26 has three significant digits, your final answer should be rounded off to three significant digits as well.

Key concepts

Understanding Powers Evaluation

Evaluating powers is a fundamental skill in mathematics, involving raising a number, known as the base, to a certain exponent. When calculating powers, the magnitude of a number changes significantly, but regardless of how large or small the resultant value is, it is crucial to maintain the correct level of precision indicated by its significant digits.

Significant digits, or significant figures, reflect the precision of a number — the more significant digits, the more precisely the number is measured or known. If a number has fewer significant digits, it implies less precision. For example, in the base number 2.26 from the exercise, there are three significant digits, which tells us that the precision of the number extends to the hundredths place.

When evaluating powers like in the expression \(2.26)^{6}\), one must be attentive to maintain that precision after performing the calculation. This ensures that the result reflects the accuracy of the original data and does not imply a false sense of precision.

Calculator Usage for Exponents

Calculators are powerful tools for performing complex mathematical operations including the evaluation of exponents. When you're tasked with raising a number to a power, like \(2.26)^{6}\), the procedure may vary slightly depending on the type of calculator you're using. However, the basic steps are generally the same.

First, you enter the base number, which in this case is 2.26. Next, you'll find the power function, often denoted by '^', 'EXP', or 'y^x'. After selecting this function, you enter the exponent value, which is 6 for this exercise. Pressing the equals button or 'Enter' will then display the calculated power.

Tip for Accurate Power Calculations

If your calculator has a 'history' or 'answer' functionality, you can use these features to retain more significant figures throughout your calculations. This can be particularly helpful for intermediate steps in longer calculations, or when you need to use the result in subsequent operations.

Rounding to Significant Figures

Rounding to significant figures is an essential step to ensure the accuracy and reliability of your results when working with numbers that have been raised to a power. After performing the calculation with a calculator, you might obtain a long string of digits. It’s important to round this result to the proper number of significant digits as dictated by the initial values in the problem.

The process of rounding can affect the final digit that is kept. For example, if a number is halfway between two possible values, you should round to the nearest even number (known as 'round half to even' or 'banker’s rounding'). This minimizes rounding errors in statistical operations. In the case of our expression \(2.26)^{6}\), after obtaining the result from the calculator, we must round it to three significant figures since the base number 2.26 has three.

Steps for Rounding

• Identify the number of significant figures needed (three, in this case).
• Count digits from the first non-zero digit.
• Round off the number at the appropriate place, taking into account the digit that follows (whether to round up or remain the same).

Remember, preserving the correct number of significant figures is a testament to the measurement's precision and should align with the initial data provided to convey the correct level of certainty in your results.