Question

Combined Operations with Approximate Numbers Perform each computation, keeping the proper number of digits in your answer. $$\sqrt[5]{(9.06)(4.86)(7.93)}$$

Short answer

The value of \(\sqrt[5]{(9.06)(4.86)(7.93)}\) rounded to three significant figures is approximately 3.48.

Step-by-step solution

Identify the most precise value

The given numbers are 9.06, 4.86, and 7.93. Take note of the number of significant figures in each value. The least precise value will determine the number of significant figures in our final answer. In this case, all values have three significant figures.

Multiply the numbers

Multiply the given numbers before applying the fifth root. Calculate the product of 9.06, 4.86, and 7.93. The product is approximately 349.090748.

Take the fifth root

Calculate the fifth root of the result from step 2, which is the product of the three numbers. Use a scientific calculator to find the fifth root of 349.090748 which is approximately 3.481519.

Round to the proper number of digits

Round the result to three significant figures because all the original values had three significant figures. The fifth root of 349.090748 rounded to three significant figures is 3.48.

Key concepts

Approximate Numbers

When working with numbers in mathematics, we often encounter values that aren't exact but are instead estimates or 'approximate numbers'. These approximations arise in measurements, statistics, and situations where exact values are either unknown or unnecessary. Approximate numbers carry with them a certain number of significant figures, which are the digits that contribute meaningfully to the measurement's precision.

For example, in the problem where we have the numbers 9.06, 4.86, and 7.93, they are all considered to be approximate because they are not whole numbers and might be results of measurements. The significance of each digit in these numbers is vital when performing arithmetic operations to ensure that the final answer represents an appropriate level of accuracy. In our example, the numbers have three significant figures each, which means when combined; the result should also reflect this degree of precision.

Combined Operations

When mathematical problems involve 'combined operations', such as multiplication followed by taking roots, as in the given exercise, the rules of significant figures become quite important. Before beginning the calculations, it's necessary to understand the precision of each number involved. Combined operations require precision because the outcome of one operation can affect the following one significantly.

In our example, multiplying 9.06, 4.86, and 7.93 is the first step. The multiplication of these numbers gives us a raw answer, but remember, this answer might contain more significant figures than our original numbers and must be managed correctly. As we proceed to the subsequent operation, which is taking the fifth root in this case, maintaining the proper level of precision ensures that our final answer is consistent with the precision of the initial values.

Rounding Numbers

Once the combined operations have been performed, we often need to 'round numbers' to express our result with the correct number of significant figures. Rounding is the process of reducing the number of digits in a number while trying to keep its value similar. The number is 'rounded' to the nearest value with the desired number of significant figures.

In the context of our exercise, after the calculation is performed and we obtain the fifth root of our product, the result carries more digits than necessary. Therefore, we round it to maintain the same level of precision as our least precise initial value. Since each of our starting numbers had three significant figures, the final result, a fifth root, should also be rounded to three significant figures, which gives us the correct answer of 3.48. This step ensures that the solution reflects the precision of the initial data, which is crucial for accurate and reliable results in any scientific or mathematical context.