Problem 6 - College Algebra

Question

Approximate each number using a calculator. Round your answer to three decimal places. \(6^{-1.2}\)

Step-by-Step Solution

Answer

The approximate value of \(6^{-1.2}\), rounded to three decimal places, is 0.068.

Step 1: Concept of Negative Power

Recall that a negative exponent means that the base is on the denominator of a fraction. Hence, instead of finding the value of \(6^{1.2}\), we are essentially finding the value of \(1/6^{1.2}\).

Step 2: Using a Calculator

Using a calculator, compute \(1/6^{1.2}\). Make sure to follow the order of operations correctly. An accurate calculator should give a precise result.

Step 3: Rounding Off

After obtaining the result, you need to round it off to three decimal places. Remember, if the digit to the right of the third decimal place is 5 or greater, round up the third digit. Otherwise, leave the third digit as it is.

Key Concepts

Calculator Usage

Using a calculator can make solving problems with negative exponents much easier and faster. Modern calculators are equipped to handle the complexities of calculations involving exponents. When you encounter a problem like calculating \(1/6^{1.2}\), one of your first steps would be to enter it correctly into your calculator. Here are some tips:

• Ensure your calculator is in the correct mode (usually 'standard' mode for basic arithmetic).
• Use the fraction key to input the division, or simply divide after finding the power.
• Double-check your exponent and make sure to input it with its proper sign.

By relying on your calculator, you reduce the risk of human error in computation, and it can give you a more accurate result quickly. Don't forget to practice entering your problem into the calculator to build speed and confidence.

Rounding Numbers

Rounding numbers is a fundamental skill when working with approximations. When you use a calculator to find values like \(1/6^{1.2}\), you may get a long decimal. However, for practical purposes, rounding to a specific number of decimal places—such as three in this exercise—makes the number more manageable. Follow these simple rules:

• Look at the digit right after your rounding point (third decimal place in this task).
• If it's 5 or higher, increase the last digit you're keeping by 1.
• If it's less than 5, keep the digit as is.

For instance, if your calculator reads 0.140702, focus on the fourth digit after the decimal point. Here, you would round 0.1407 to 0.141 as the digit '7' prompts a rounding up.Rounding ensures ease of use, accuracy in presentations, and is an important step in many mathematical problems.

Order of Operations

Understanding the order of operations is crucial in accurately solving problems like \(1/6^{1.2}\). The order of operations is the rule that dictates the correct sequence to follow when performing calculations. The common acronym is PEMDAS:

• P for Parentheses
• E for Exponents
• M for Multiplication
• D for Division
• A for Addition
• S for Subtraction

When calculating expressions like \(1/6^{1.2}\), you first deal with the exponent, calculating \(6^{1.2}\). Then, you take the reciprocal, which is equivalent to dividing 1 by the result of the exponentiation. By following these steps correctly, you avoid errors, making sure that the expression is solved properly before moving to rounding. This concept is especially important when using a calculator, as it needs explicit instructions that align with the order of operations to produce the correct result.