Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Applications of Roots. The geometric mean \(B\) between 3.75 and 9.83 is $$B=\sqrt{(3.75)(9.83)}$$ Evaluate \(B\)

Short answer

The geometric mean B is approximately 6.07, rounded to three significant digits.

Step-by-step solution

Understand Geometric Mean

The geometric mean between two numbers is the square root of the product of those numbers. It gives the central tendency or typical value of a set of numbers by using the product of their values.

Multiply the Numbers

Multiply 3.75 by 9.83 to get their product. Calculate the product before taking the square root.

Take the Square Root

After finding the product, take the square root of that number to find the geometric mean, which is B.

Retain Significant Digits

The number with the least number of significant digits is 3.75 with three significant digits. Therefore, our answer should also be rounded to three significant digits.

Calculate and Round

Perform the calculation and round the result to three significant digits. $$ B = \sqrt{(3.75)(9.83)} \approx \sqrt{36.8625} \approx 6.07 $$ (rounded to three significant digits).

Key concepts

Significant Digits

Understanding significant digits is crucial when reporting a measurement or result in mathematics and science. Significant digits, also known as significant figures, represent the number of meaningful digits in a number, starting with the first non-zero digit. For instance, in the number 3.75, all three digits are significant because they contribute to the precision of that value.

When performing calculations, the rule is to report the final answer with the same number of significant digits as the number with the least significant digits used in the calculation. This maintains the precision of the result. If we have the number 9.83 with three significant digits and 3.75 also with three significant digits, our final answer must be rounded to three significant digits to accurately reflect the precision of our measurements. It is a subtle yet important principle that helps maintain the integrity of numerical computations.

Square Roots

The square root is a fundamental concept in mathematics, symbolized by \( \sqrt{x} \). It represents a value that, when multiplied by itself, gives the original number. The square root of a number can be thought of as the opposite of squaring a number. For example, since \( 7 \times 7 = 49 \), the square root of 49 is 7, expressed as \( \sqrt{49} = 7 \).

In the context of calculating geometric means, the square root allows us to find the number that evenly spans between two values. When we compute the geometric mean, we multiply the values together and take the square root of the product. This principle is essential in understanding geometric mean as a form of central tendency.

Central Tendency

Central tendency is a statistical measure that identifies the center or typical value of a dataset. There are several measures of central tendency, with mean, median, and mode being the most common. The geometric mean, which we encounter in this exercise, is particularly useful when dealing with proportional growth, such as interest rates or population growth.

Geometric mean is found by multiplying all the numbers in a set together and then taking the nth root (where n is the number of values in the set). In the case of two values, this becomes the square root of the product. The geometric mean is a type of average that can give a more representative central value when dealing with datasets that contain very high or very low numbers, as it tends to dampen the effect of extreme values.

Rounding Numbers

Rounding numbers is a method used to reduce the digits in a number while keeping its value similar to what it was originally. This technique is especially important after calculations that result in long decimal numbers or when we need to express a result to a certain level of precision, based on significant digits. The process involves looking at the digit immediately after the last digit we want to keep.

If this digit is five or higher, we increase the last desired digit by one (this is called rounding up). If it's lower than five, we leave the last desired digit as it is (rounding down). In our geometric mean calculation, after finding our product, we might end up with a long decimal, but we round it to three significant digits, matching the least precision from our original values.