Question

Evaluate the expression using a calculator. Round your answer to two decimal places when appropriate. (See Example 3.) \(86^{-5 / 6}\)

Short answer

After calculating, \(86^{-5 / 6}\) equals approximately 0.16 when rounded to two decimal places.

Step-by-step solution

Input the expression into the calculator

Begin by inputting the expression \(86^{-5 / 6}\) into the calculator. Make sure to input the correct order of numbers and symbols into the calculator.

Calculate the expression

After inputting the expression into the calculator correctly, press the equal button to calculate. You should get a decimal number as your result.

Round the answer

The last step is rounding the result to the nearest two decimal places. If the third decimal place is a 5 or more, round up. If it is less than 5, round down.

Key concepts

Rounding

Rounding numbers is about making a number simpler but keeping its value close to what it was. It's like cutting some hair but keeping the style. When rounding to two decimal places, look at the third number after the decimal point. This will tell you if you should round up or down.
If the third number is 5 or more, give the second decimal number a little push up. For example, if you have 3.456, after rounding, it becomes 3.46.
But if the third number is less than 5, keep the second number the same. So, 3.432 would stay as 3.43 after rounding.
Think of rounding like a helpful tool to make numbers easier to read, especially when the exact number is not needed. It's like saying "around 20" instead of "19.7645". Always remember to round numbers only when specified.

Calculator Skills

Using a calculator correctly is an essential skill for students when solving complex mathematical expressions. It’s not just about pressing buttons; it’s about knowing which buttons to press and in what order. When you have an expression, such as \(86^{-5 / 6}\), it is crucial to input it carefully.
Make sure to follow these general steps:

• Identify the mathematical operation - in this case, it's working with exponents.
• Enter the numbers as they appear in the expression. Use parentheses if necessary to maintain the correct order of operations.
• Check if your calculator is in the right mode for the calculation - sometimes, calculators have settings for degrees or radians, which don't apply here, but it's good to know your device's settings.

Once the number is entered, simply press the '=' button to get your solution. Double-check your entry to avoid any surprises and ensure that you’ve respected the order of operations.
Calculators help speed up calculations and minimize errors when used correctly. They're like a trusty math partner ready to assist whenever needed.

Negative exponents

Negative exponents might seem tricky, but with a little practice, they become easy to manage. A negative exponent indicates that you should take the reciprocal of the base number. In simpler terms, you flip the number. For example, \(x^{-n} = \frac{1}{x^n}\).
For the expression \(86^{-5/6}\), it signifies \(\frac{1}{86^{5/6}}\). This means you first calculate \(86^{5/6}\) and then take the reciprocal.
When dealing with negative exponents, remember:

• Negative exponents mean 'divide by that power', not multiplication.
• Flip the base to switch the sign of the exponent.
• Perform the operations exactly like positive exponents after the reciprocal step.

Understanding negative exponents is crucial because they frequently appear in various mathematical problems and real-world situations like calculating decay or odd-numbered roots. With regular practice, they will soon feel as familiar as positive exponents.