# Calculate the following: 1. Geometric mean of sqrt17 + sqrt13 and sqrt17 -sqrt13 2. (2/7)*(14/5) + (1/5)

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To calculate

1. Geometric mean of sqrt17 + sqrt13 and sqrt17 -sqrt13

WE know GM of xand y is (xy)^(1/2).

Therefore GM of sqrt17+sqrt13 and sqrt17-sqrt13 is{ (sqrt17+sqrt13){sqrt17-sqrt13)} ^(1/2) = {17-13}^(1/2) = 4^(1/2) = **2**, as (a+b)(a-b) = a^2-b^2.

2)

(2/7)*(14/5) + (1/5)

(2/7)(14/5) = 2*14/(7*5) = 4/5.

4/5 +1/5 = 5/5 = 1.

So (2/7)(14/5)+1/5 = 1.

The geometric mean of A and B is sqrt (A*B).

Here A is sqrt17 + sqrt13 and B is sqrt17 -sqrt13.

A*B = ( sqrt17 + sqrt13 ) * ( sqrt17 -sqrt13 )

= ( sqrt 17 )^2- ( sqrt 13)^2

= 17 -13

= 4

sqrt 4 = 2

**Therefore the geometric mean is 2**

(2/7)*(14/5) + (1/5)

= ( 2*14)/ ( 7*5) + ( 1/5)

= 2 * 2 / 5 + 1/5

= 4/5 + 1/5

= 5/5

=1

**(2/7)*(14/5) + (1/5) = 1**

1. To determine the geometric mean of 2 numbers, a and b, we'll apply the formula:

G.m. = sqrt(a*b)

We'll identify a and b.

a = sqrt17 + sqrt13

b = sqrt17 - sqrt13

a*b = (sqrt17 + sqrt13)(sqrt17 - sqrt13)

The product is the result of the difference of squares:

(a-b)(a+b) = a^2 - b^2

(sqrt17 + sqrt13)(sqrt17 - sqrt13) = (sqrt17)^2 - (sqrt13)^2

(sqrt17 + sqrt13)(sqrt17 - sqrt13) = 17 - 13

(sqrt17 + sqrt13)(sqrt17 - sqrt13) = 4

Now, we'll calculate the geometric mean:

G.m = sqrt 4

**G.m. = 2**

2. To calculate the expression, we'll keep the algebraic operations order. So, we'll calculate the multiplication and after that the addition.

The multiplication of the 2 ratios:

(2/7)*(14/5) = 2*2*7/7*5

We'll simplify and we'll get:

(2/7)*(14/5) = 4/5

Now, we'll calculate the addition:

(4/5) + (1/5)

Since the 2 ratios have the same denominator, we'll add the numerators:

(4/5) + (1/5) = (4+1)/5

(4/5) + (1/5) = 5/5

(4/5) + (1/5) = 1

The simplified result of the expression is:

**(2/7)*(14/5) + (1/5) = 1**