Question

Simplify - write as a product - compute - 100 points

Answer 1

The answer I got was -1

Answer 2

a)

`\sqrt{61 - 24 √(5) } = - 4 + 3 √(5)`

b)

`( \sqrt{ ( {c}^(2) - 1) ({b}^(2) - 1) } - {2 √(bc) }) (\sqrt{ ( {c}^(2) - 1) ({b}^(2) - 1) } + {2 √(bc) } )`

c)

`\frac{ \sqrt{9 - 4 √(5) } }{2 - √(5) } = - 1`

Explanation:

We want to simplify

`\sqrt{61 - 24 √(5) }`

Let :

`\sqrt{61 - 24 √(5) } = a - b √(5)`

Square both sides:

`(\sqrt{61 - 24 √(5) } )^(2) = ({a - b √(5) })^(2)`

Expand;

`61 - 24 √(5) = {a}^(2) - 2ab √(5) + 5 {b}^(2)`

Compare coefficient:

`{a}^(2) + 5 {b}^(2) = 61 - - - (1)`

`- 24 = - 2ab \\ ab = 12 \\ b = (12)/(b) - - -( 2)`

Solve simultaneously,

`\frac{144}{ {b}^(2) } + 5 {b}^(2) = 61`

`5 {b}^(4) - 61 {b}^(2) + 144 = 0`

Solve the quadratic equation in b²

`{b}^(2) = 9 \: or \: {b}^(2) = (16)/(5)`

This implies that:

`b = \pm3 \: or \: b = \pm (4 √(5) )/(5)`

When b=-3,

`a = - 4`

Therefore

`\sqrt{61 - 24 √(5) } = - 4 + 3 √(5)`

We want to rewrite as a product:

`{b}^(2) {c}^(2) - 4bc - {b}^(2) - {c}^(2) + 1`

as a product:

We rearrange to get:

`{b}^(2) {c}^(2) - {b}^(2) - {c}^(2) + 1- 4bc`

We factor to get:

`{b}^(2) ( {c}^(2) - 1) - ({c}^(2) - 1)- 4bc`

Factor again to get;

`( {c}^(2) - 1) ({b}^(2) - 1)- 4bc`

We rewrite as difference of two squares:

`(\sqrt{( {c}^(2) - 1) ({b}^(2) - 1) })^(2) - ( {2 √(bc) })^(2)`

We factor further to get;

`( \sqrt{ ( {c}^(2) - 1) ({b}^(2) - 1) } - {2 √(bc) }) (\sqrt{ ( {c}^(2) - 1) ({b}^(2) - 1) } + {2 √(bc) } )`

c) We want to compute:

`\frac{ \sqrt{9 - 4 √(5) } }{2 - √(5) }`

Let the numerator,

`\sqrt{9 - 4 √(5) } = a - b √(5)`

Square both sides;

`9 - 4 √(5) = {a}^(2) - 2ab √(5) + 5 {b}^(2)`

Compare coefficients;

`{a}^(2) + 5 {b}^(2) = 9 - - - (1)`

and

`- 2ab = - 4 \\ ab = 2 \\ a = (2)/(b) - - - - (2)`

Put equation (2) in (1) and solve;

`\frac{4}{ {b}^(2) } + 5 {b}^(2) = 9`

`5 {b}^(4) - 9 {b}^(2) + 4 = 0`

`b = \pm1`

When b=-1, a=-2

This means that:

`\sqrt{9 - 4 √(5) } = - 2 + √(5)`

This implies that:

`\frac{ \sqrt{9 - 4 √(5) } }{2 - √(5) } = ( - 2 + √(5) )/(2 - √(5) ) = ( - (2 - √(5)) )/(2 - √(5) ) = - 1`