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Chapter 5: Problem 21

Find the geometric mean between 4 and 36

Short Answer

Expert verified
The geometric mean between 4 and 36 is 12.

Step by step solution

01

- Understand the Formula for Geometric Mean

The geometric mean of two numbers a and b is given by the formula: \ \( \sqrt{a \cdot b} \)
02

- Substitute the Numbers into the Formula

Substitute the given numbers 4 and 36 into the formula: \ \( \sqrt{4 \cdot 36} \)
03

- Multiply the Numbers

Calculate the product of 4 and 36: \ \( 4 \cdot 36 = 144 \)
04

- Find the Square Root

Find the square root of 144: \ \( \sqrt{144} = 12 \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

geometric mean formula
The concept of the geometric mean is a bit different from the more familiar arithmetic mean. Whereas the arithmetic mean is all about adding and dividing, the geometric mean uses multiplication and square roots.
The geometric mean between two numbers, say a and b, is found using this formula: \[ \sqrt{a \cdot b} \] This means you'll multiply the two numbers and then take the square root of that product.
Let's break that down:
  • First, you find the product of the numbers a and b.
  • Next, you find the square root of their product.
It's a handy way to find the average value in terms of multiplication, particularly useful in geometry and growth rates. Now, with this understanding, let's move on to applying it step by step.
finding square root
To find the geometric mean, you'll need to find the square root of the product of two numbers. The square root is a value that, when multiplied by itself, gives the original number.
For example, the square root of 144 is 12, because 12 times 12 equals 144.
Here鈥檚 how you can find the square root:
  • Identify the number whose square root you need to find.
  • Use either a calculator or estimate manually by finding a number that, when multiplied by itself, gives the original number.
  • If using a calculator, simply enter the product and press the square root button (鈭�).
Finding the square root is especially important for the geometric mean because it completes our calculation to find the average value of the numbers involved.
multiplication of numbers
Another crucial step in calculating the geometric mean involves multiplying numbers.
In our example, we're finding the geometric mean of 4 and 36.
Here鈥檚 the multiplication process broken down:
  • Start by writing down the numbers you need to multiply: 4 and 36.
  • Multiply these numbers: 4 * 36.
  • The result of this multiplication is 144.

This step is necessary because it forms the basis of what you鈥檒l take the square root of in order to find the geometric mean.
Practice and familiarity with multiplication will make this step quicker and easier. Multiplication and finding square roots go hand in hand in many mathematical calculations, especially with averages like the geometric mean.

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