Question

Use a calculator to evaluate each expression. Round your answer to three decimal places. \((6.1)^{-3}\)

Short answer

0.004

Step-by-step solution

Understand the Expression

We need to evaluate the expression \(6.1^{-3}\) using a calculator and round the result to three decimal places.

Use the Calculator

Input the expression \(6.1^{-3}\) into the calculator. To do this, you can use the exponentiation function found on most scientific calculators.

Calculate the Result

Perform the calculation. The calculator will display a value which you will need to round to three decimal places.

Round the Result

Round the result obtained from the calculator to three decimal places. If the fourth decimal is 5 or greater, round the third decimal up by one; if it is less than 5, keep the third decimal as is.

Write the Final Answer

Write down the rounded number as your final answer.

Key concepts

Exponentiation

Exponentiation is a fundamental mathematical operation where a number, called the base, is raised to the power of an exponent. In the expression \(6.1^{-3}\), 6.1 is the base and -3 is the exponent. To calculate this, input the base and exponent into your scientific calculator using the exponentiation function, often labeled as \(y^x\) or \(^ \).
When the exponent is negative, it implies that we are taking the reciprocal of the base raised to the corresponding positive exponent. For example, \(6.1^{-3}\) is equivalent to \( \frac{1}{{6.1^3}} \). This means you first calculate \(6.1^3 \) and then take its reciprocal.

Rounding Decimals

After you have calculated the value using your calculator, the next step is to round the result to three decimal places.
Rounding is the process of adjusting a number to make it simpler but keeping it close to the original value. To round to three decimal places, follow these steps:

• Identify the digit in the fourth decimal place.
• If this digit is 5 or greater, you round up the third decimal place by one.
• If this digit is less than 5, you leave the third decimal place unchanged.

Let's say your calculator shows a result of 0.004132. To round this to three decimal places, look at the fourth decimal digit (3 in this case). Since it is less than 5, the rounded result would be 0.004. On the other hand, if the result was 0.004157, because the fourth decimal digit (5) is 5 or greater, it would be rounded up to 0.005.

Negative Exponents

Negative exponents can be a bit tricky at first. They denote the reciprocal of the base raised to the corresponding positive exponent. For instance, \(a^{-n}\) is equivalent to \( \frac{1}{{a^n}} \).
This concept is crucial when evaluating expressions involving negative exponents.
When faced with \(6.1^{-3} \), you are essentially calculating 1 divided by \(6.1^3 \). Performing \(6.1 \) raised to the power of 3 gives you a large number, and taking its reciprocal makes it much smaller.
Understanding how negative exponents work will help you tackle a wide range of problems involving exponentiation efficiently and accurately.