how to find the geometric mean of two numbers

Tabular array of Contents

1. Geometric Mean Definition
• Geometric Mean vs Arithmetic Mean
2. How To Discover Geometric Mean
3. Geometric Mean Formula
4. Geometric Hateful Uses
5. Geometric Mean Examples

Geometric Mean Definition

Geometric mean involves roots and multiplication, non addition and division. Y'all get geometric mean by multiplying numbers together and then finding the ${\mathrm{northward}}^{th}$ root of the numbers such that the $n^{th}$ root is equal to the corporeality of numbers you multiplied. Geometric mean is useful in many circumstances, especially problems involving money.

The geometric mean is the $n^{th}$ root when you multiply $n$ numbers.

For example, if y'all multiply three numbers, the geometric hateful is the third root of the product of those three numbers. The geometric mean of five numbers is the fifth root of their product.

Suppose we said we found the geometric mean using the ${11}^{th}$ root of the numbers. That tells you that 11 numbers were multiplied together. To detect the geometric mean of 4 numbers, what root would we take? The fourth root, of grade.

Geometric Hateful vs Arithmetic Mean

Y'all are probably familiar with arithmetic mean, informally chosen the boilerplate of a group of numbers. You get arithmetic mean by arithmetics, or adding the numbers together and so dividing past the amount of numbers you were adding.

How to Find the Geometric Mean

Nosotros will start with an easy instance using only two numbers, 4 and 9. What is the geometric mean of 4 and nine?

Multiply $\mathrm{four}\times 9$. So find the foursquare root of their product (because yous simply multiplied two numbers):

$\mathrm{iv}\times 9=36$

$\sqrt{36}=\mathrm{half\; dozen}$

The geometric mean of 4 and 9 is 6.

Geometric Hateful Formula

Let $\mathrm{northward}$ equal the number of terms nosotros are multiplying, and allow ${\mathrm{10}}_1$, $x_2$, ${\mathrm{ten}}_3$ and and so on up to ${\mathrm{ten}}_n$ exist the unlike factors (the various terms).

Geometric Mean Theorem

This formula tells the states to multiply all the terms (radicands) within the radical (the symbol for roots), and and so to find the $n^{th}$ root of them where $n$ is how many radicands you accept. You lot can divide whole number radicands with either an $\times$ or a $*$ to show you are multiplying them.

Permit's kickoff try it with our earlier, easy example, and here the $\times$ is the symbol of multiplication:

$\sqrt[2]{\mathrm{iv}\times 9}$

Nosotros can substitute a $*$ for the $\times$ to also show multiplication:

$\sqrt[2]{\mathrm{four}*9}$

Now let's effort a quick example with three terms:

$\sqrt[\mathrm{iii}]{3*\mathrm{vi}*12}$

The product of three x half dozen 10 12 = 216.

$\sqrt[3]{216}$

The cube root (the third root) of 216 is 6.

Our geometric mean is 6.

Uses for the Geometric Mean

Someday we are trying to calculate boilerplate rates of growth where growth is determined by multiplication, not addition, we need the geometric mean. This connects geometric mean to economic science, financial transactions between banks and countries, interest rates, and personal finances.

Your growth rate for coin you have in bank deposits can be calculated using geometric mean, since your coin grows at an advertised charge per unit. Yous could non calculate this using arithmetic hateful.

Geometric Mean Examples

The all-time way to become familiar with using the geometric mean is to employ it. Utilize the formula to find the geometric mean of these 6 numbers:

$\mathrm{ii},3,\mathrm{v},\mathrm{three},10,8$

Hither is the formula again:

$\sqrt[n]{x_{\mathrm{i}}\times x_2\times x_3\cdots {\mathrm{10}}_{\mathrm{due\; north}}}$

And here is the formula with our numbers:

$\sqrt[\mathrm{vi}]{2\times 3\times 5\times 3\times \mathrm{x}\times 8}$

The product of the radicands is found easily:

$(2\times 3)(\mathrm{v}\times 3)(10\times 8)=?$

$6\times 15\times 80=?$

$90\times \mathrm{eighty}=\mathrm{7,200}$

Now you must detect the sixth root of 7,200:

$\sqrt[\mathrm{half\; dozen}]{\mathrm{vii,200}}$

The sixth root of $\mathrm{7,200}=4.39429035137$. Our work is done!

Lesson Summary

In this lesson nosotros learned how to define the geometric mean, which is the ${\mathrm{northward}}^{th}$ root of a group of $n$ factors, how to observe the geometric hateful of whatsoever group of numbers by multiplying them and so taking the root equal to the total corporeality of numbers, how to utilize the geometric mean to situations where growth rates are adamant by multiplication, non addition, and how to write and use the formula for geometric mean.

Next Lesson:

Angle Bisector Theorem

Source: https://tutors.com/math-tutors/geometry-help/geometric-mean

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