Radicals

Review

In the study of right triangles we will be working with radicals. Below is a quick review to refresh your understanding of radicals and allow you to simplify and work with radicals.

Properties of Radicals

	The square root of product equals the product of the square roots of the factors	Product Property
	The square root of a quotient equals the square roots of the numerator and denominator	Quotient Property

In the expression is a radical and a is called the radicand.

A radical expression is in its simplest form if :

No radicands have perfect square factors other than 1
No radicands are fractions
No Radicals are in the denominator of a fraction

If the denominator of the fraction has a radicand that is not a perfect square we multiply the numerator and denominator by a radical that will make the denominator a perfect square:

We multiplied the numerator and denominator by

1. 2. 3. 4.

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Geometric Mean

We are all familiar with the arithmetic mean of two numbers: It is the number which would be exactly between the two given numbers if they were on a number line. For example 10 is the arithmetic mean of 4 and 16:

We see that that the arithmetic mean is the number such that the difference or distance between all three numbers is the same, 10 - 4 = 6 and 16 - 10 = 6. Thus 10 is 6 away from 4 and 6 away from 16. So this leads us to a formula for finding the arithmetic mean. Let x be the arithmetic mean between to numbers a and b where b > a. We would have x - a = b - x, or 2x = b - a, so

x = (b+a)/2 arithmetic mean

The geometric mean is similar to the arithmetic mean, but the geometric mean of two numbers is the number such that the ratios between the three numbers is the same. Let's consider our previous example of 4 and 16. The geometric mean between two positive numbers is a positive number such that when we divide the geometric mean by 4 it is gives us the same result as when we divide 16 by the geometric mean.

Practice:

1. How would you set up the problem to find the geometric mean of 4 and 16? Write the problem down on your practice paper.

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2. Now solve for the geometric mean.

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3. Let a and b be two positive numbers such that a < b. Write an equation for the geometric mean, x, in terms of a and b.

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The geometric mean between two positive numbers a,b is the number x such that

Click the light bulb to practice what you have learned.