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Chapter 5: Problem 23

Find the geometric mean between 9 and 64.

Short Answer

Expert verified
The geometric mean between 9 and 64 is 24.

Step by step solution

01

Understand the Geometric Mean

The geometric mean of two numbers is found by multiplying them together and then taking the square root of the product.
02

Multiply the Numbers

Multiply the two numbers, 9 and 64, together: \[ 9 \times 64 = 576 \]
03

Take the Square Root

Take the square root of the product obtained in Step 2: \[ \text{Geometric Mean} = \sqrt{576} = 24 \]

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Key Concepts

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

Multiplication
In order to find the geometric mean, you first need to multiply the numbers together. Multiplication is the process of adding a number to itself a certain number of times. Let's look at an example to understand better. If you have two numbers, 9 and 64, and you need to find the geometric mean between them, the first step is to multiply them together. In mathematical terms, this is represented as:
\( 9 \times 64 = 576 \)
Multiplication is straightforward when you remember: One number is repeated as many times as the value of the other number. Always ensure your basic multiplication skills are solid, as this step is crucial for calculating the geometric mean.
Square Root
The square root is essential for finding the geometric mean. In essence, the square root of a number is a value that, when multiplied by itself, gives the original number. For instance, in our geometric mean problem, after obtaining 576 from multiplying 9 and 64, the next step is to take the square root of 576. This can be represented as:
\[ \sqrt{576} = 24 \]
So, the square root of 576 is 24 because 24 multiplied by itself equals 576. Square roots help in simplifying the problem and giving us the required result. You just have to find a value that, when squared, brings you back to your original number.
Problem-Solving Steps
Solving for the geometric mean involves a set of clear steps. Let's break them down:
  • Step 1: Understand the concept. The geometric mean of two numbers is found by taking their product and then finding the square root of that product. This means you have to use both multiplication and square roots.

  • Step 2: Multiply the numbers. In our example, multiply 9 and 64. The multiplication gives us 576.

  • Step 3: Find the square root. The geometric mean is found by taking the square root of the product. Here, we take the square root of 576, which is 24.


Each of these steps is straightforward but needs to be followed accurately to obtain the correct answer. If each step is performed correctly, solving for the geometric mean becomes a simple process.

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