Given the numbers a=square root(5-square root21) and b=square root(5+square root21), what is the product of a^-1*b^-1?

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

It is given that a = sqrt(5 - sqrt 21) and b = sqrt(5 + sqrt 21)

We have to determine a^(-1) * b^(-1) = 1/(a*b)

a*b = [sqrt(5 - sqrt 21)][sqrt(5 + sqrt 21)]

=> sqrt [(5 - sqrt 21)(5 + sqrt 21)]

=> sqrt [ 5^2 - (sqrt 21)^2]

=> sqrt [25 - 21]

=> sqrt 4

=> +/-2

1/(a*b) = +/-1/2

The product a^-1*b^-1 is 1/2 or -1/2

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll use the negative power rule:

a^-1 = 1/a

b^-1 = 1/b

(a^-1)*(b^-1) = 1/a*b

1/a*b = 1/sqrt[(5-sqrt21)*(5+sqrt21)]

We'll transform the product of radicands into a difference of 2 squares, using the formula:

(x-y)(x+y) = x^2 - y^2

1/a*b = 1/sqrt(5^2 - 21)

1/a*b =1/sqrt(25-21)

1/a*b = 1/sqrt4

1/a*b = 1/2 or 1/a*b = -1/2

The possible values of the product (a^-1)*(b^-1) are: {-1/2 ; 1/2}.

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