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Chapter 9: Problem 47

Find the geometric mean of 12 and 48 .

Short Answer

Expert verified
The geometric mean is 24.

Step by step solution

01

Understanding the Geometric Mean Formula

The geometric mean of two numbers is calculated by taking the square root of the product of those numbers. If the two numbers are \(a\) and \(b\), the geometric mean \(G\) is given by the formula: \[ G = \sqrt{a \times b} \]
02

Calculating the Product of Given Numbers

Identify the two numbers given: 12 and 48. Calculate their product: \[ 12 \times 48 = 576 \]
03

Taking the Square Root

Now, take the square root of the product obtained in Step 2:\[ G = \sqrt{576} \]
04

Computing the Square Root

Compute the square root of 576. Since 576 is a perfect square, its square root is 24. Thus, the geometric mean of 12 and 48 is:\[ G = 24 \]

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Key Concepts

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

Understanding Square Roots
The square root is a fundamental concept in mathematics, often symbolized by the radical sign (鈭�). It refers to a number which, when multiplied by itself, returns the original value. For instance, the square root of 25 is 5 because 5 multiplied by 5 equals 25.
In formula format, if you have a number \( x \), its square root is \( \sqrt{x} \).
  • The square root operation is directly relevant when finding the geometric mean.
  • It is especially convenient when we calculate the geometric mean, which is the square root of the product of numbers.
In the context of the original exercise, we found the square root of 576, yielding 24.
Defining Perfect Squares
A perfect square is a number that can be expressed as the product of an integer with itself. This means that when an integer is squared, it results in a perfect square. Examples include 1, 4, 9, 16, and 25.
Identifying a number as a perfect square simplifies finding its square root considerably, as it will always be an integer.
  • In our original problem, 576 is identified as a perfect square.
  • Recognizing perfect squares is useful in quickly determining their square roots without complex calculations.
Thus, knowing 576 as a perfect square made obtaining its square root of 24 much easier.
Exploring the Product of Numbers
The product of numbers is the result you get when you multiply those numbers together. This operation is crucial for numerous mathematical concepts, including the calculation of the geometric mean.
When dealing with numbers in geometry, the product often precedes taking another operation, like square roots.
  • The mathematical expression for the product of two numbers \( a \) and \( b \) is \( a \times b \).
  • In finding the geometric mean, you first need to calculate this product.
In our example, multiplying 12 and 48 gave us 576, setting the stage for the next steps in finding the geometric mean.

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