Problem 50 - Algebra 1

Question

Use a calculator to evaluate the expression. Round your answer to the nearest hundred thousandth. $$(1.1)^{-2}$$

Step-by-Step Solution

Answer

The expression \((1.1)^{-2}\) evaluates to approximately 0.826446 when rounded to the nearest hundred thousandth.

Step 1: Raise 1.1 to the 2nd power

Use your calculator to square the number 1.1. You should get a result of 1.21.

Step 2: Take the reciprocal of the result

Now take the reciprocal of 1.21 which means divide 1 by 1.21. This would give an answer of approximately 0.826446281.

Step 3: Round to the nearest hundred thousandth

As per the question, round the result to the nearest hundred thousandth. The hundred thousandths place is the 6th digit after the decimal. So, the result is rounded off to 0.826446.

Key Concepts

Rounding Numbers

Rounding numbers is a process that helps simplify numbers for easier interpretation or calculation. When rounding to the nearest hundred thousandth, you focus on the sixth digit after the decimal point. For example, if you have a number like 0.826446281, you look at the sixth digit, which here is '6'. If this digit is 5 or more, you round up, increasing the last rounded digit. If it is less than 5, you round down by leaving the digit as is. In our example, since the sixth number is '6', we round up, resulting in 0.826446.

Reciprocals

A reciprocal is a mathematical term used to describe the inverse of a number. To find the reciprocal, you divide 1 by the number. This flips the number's position relative to 1. For instance, in our problem, the reciprocal of 1.21 is found by calculating \( \frac{1}{1.21} \). This results in approximately 0.826446281. Reciprocal concepts are highly useful in algebra and calculus, often helping in solving equations and dealing with division of fractions.

Calculator Use

A calculator is an essential tool for evaluating mathematical expressions quickly and accurately. When dealing with complicated calculations like exponentiation or reciprocals, a calculator aids in delivering precise results. To use a calculator for these tasks:

• Enter the base number (e.g., 1.1 for our problem).
• Use the exponentiation feature to square it (usually marked as ^ or x²), obtaining 1.21.
• Next, for reciprocals, use 1 ÷ [value] for division, hence 1 ÷ 1.21, to receive a reciprocal of approximately 0.826446281.
• Lastly, ensure your calculator is set to display enough decimal places for rounding, such as to the hundred thousandth, ensuring accuracy in results.

The calculator speeds up computations and reduces human error, making it indispensable for learning and applying mathematical concepts.